diff --git a/crates/clawhdf5-format/src/storage.rs b/crates/clawhdf5-format/src/storage.rs
index 5fe9497..6b74378 100644
--- a/crates/clawhdf5-format/src/storage.rs
+++ b/crates/clawhdf5-format/src/storage.rs
@@ -89,6 +89,21 @@ pub trait Storage {
     fn as_contiguous(&self) -> Option<&[u8]> {
         None
     }
+
+    /// A hint that `[offset, offset + len)` is about to be read by the
+    /// same operation: a parser that has just learnt where the next
+    /// structures are (a node's children, a structure's body) says so
+    /// before it reads them one at a time. Nothing is read and nothing
+    /// fails. The default ignores it, as does every backend that reads
+    /// when asked; the browser's restartable reader, which fetches over
+    /// the network between attempts, fetches hinted bytes along with the
+    /// bytes an attempt actually missed, so structures a parser only
+    /// reaches after a miss arrive in the same round trip. Results never
+    /// depend on hints.
+    #[inline]
+    fn hint(&self, offset: u64, len: usize) {
+        let _ = (offset, len);
+    }
 }
 
 impl Storage for [u8] {
@@ -148,6 +163,11 @@ impl<T: Storage + ?Sized> Storage for &T {
     fn as_contiguous(&self) -> Option<&[u8]> {
         (**self).as_contiguous()
     }
+
+    #[inline]
+    fn hint(&self, offset: u64, len: usize) {
+        (**self).hint(offset, len)
+    }
 }
 
 impl<T: Storage + ?Sized> Storage for Box<T> {
@@ -170,6 +190,11 @@ impl<T: Storage + ?Sized> Storage for Box<T> {
     fn as_contiguous(&self) -> Option<&[u8]> {
         (**self).as_contiguous()
     }
+
+    #[inline]
+    fn hint(&self, offset: u64, len: usize) {
+        (**self).hint(offset, len)
+    }
 }
 
 #[cfg(feature = "std")]
@@ -193,6 +218,11 @@ impl<T: Storage + ?Sized> Storage for std::sync::Arc<T> {
     fn as_contiguous(&self) -> Option<&[u8]> {
         (**self).as_contiguous()
     }
+
+    #[inline]
+    fn hint(&self, offset: u64, len: usize) {
+        (**self).hint(offset, len)
+    }
 }
 
 /// `storage.len()` as the `usize` the parsers' end-of-file errors report
diff --git a/crates/clawhdf5-wasm/src/lazy.rs b/crates/clawhdf5-wasm/src/lazy.rs
index dadb939..e45dce4 100644
--- a/crates/clawhdf5-wasm/src/lazy.rs
+++ b/crates/clawhdf5-wasm/src/lazy.rs
@@ -25,6 +25,14 @@
 //! pass per block it needs. In practice it is one pass per *wave* of
 //! misses: a chunked read asks for all the chunks of a batch at once.
 //!
+//! Parsers also say what they are about to read ([`Storage::hint`]: a
+//! node's children, a structure's body, a listed group's child headers).
+//! A pass that misses fetches the hinted blocks it lacks too, as far as the
+//! operation's fetch budget allows, so what the parser would only have
+//! reached on the next pass arrives in the same round trip; a pass that
+//! misses nothing ignores its hints, so they never add a round trip, and
+//! results never depend on them.
+//!
 //! Blocks are kept in an LRU cache with a byte budget, trimmed only when no
 //! operation is in flight. Blocks fetched for bulk reads (raw data: a
 //! `read_ranges` call, or a read longer than a block) go first, so reading
@@ -50,6 +58,14 @@ pub const DEFAULT_MAX_FETCH: u64 = 512 << 20;
 /// the caller of [`LazyStorage::attempt`]: a pass that missed is re-run.
 pub const NEED_BYTES: &str = "bytes not fetched yet (restartable read)";
 
+/// Most blocks one pass records as hinted (see [`Storage::hint`]).
+const MAX_HINTED: usize = 1 << 16;
+
+/// Total length of `ranges`.
+fn ranges_len(ranges: &[Range<u64>]) -> u64 {
+    ranges.iter().map(|r| r.end - r.start).sum()
+}
+
 /// Settings of a [`LazyStorage`].
 #[derive(Debug, Clone, PartialEq, Eq)]
 pub struct LazyConfig {
@@ -94,6 +110,9 @@ pub struct LazyStats {
     pub bytes_fetched: u64,
     /// Blocks evicted to stay within the budget.
     pub evictions: u64,
+    /// Blocks asked for because a parser hinted it would read them
+    /// ([`Storage::hint`]), not because a pass missed them.
+    pub hinted_blocks: u64,
     /// Bytes cached now.
     pub cached_bytes: u64,
 }
@@ -125,6 +144,9 @@ struct State {
     /// Blocks the current pass missed, and whether a small read wanted
     /// them (metadata).
     missing: HashMap<u64, bool>,
+    /// Blocks the current pass was told it is about to read and that are
+    /// not cached ([`Storage::hint`]), in file order.
+    hinted: BTreeSet<u64>,
     /// Blocks a bulk read missed that have not been supplied yet: kept
     /// as bulk when they arrive.
     bulk_pending: BTreeSet<u64>,
@@ -155,6 +177,19 @@ pub struct Operation<'a> {
 }
 
 impl Operation<'_> {
+    /// One pass of `f`, as [`LazyStorage::attempt`], but a pass that misses
+    /// also asks for the blocks it was hinted it would read, as many as fit
+    /// in what is left of the operation's budget ([`LazyConfig::max_fetch`])
+    /// after the blocks it missed.
+    pub fn attempt<T>(&self, f: impl FnOnce() -> T) -> Step<T> {
+        let left = self
+            .storage
+            .config
+            .max_fetch
+            .saturating_sub(self.fetched.get());
+        self.storage.attempt_within(f, left)
+    }
+
     /// Count `ranges` against the operation's budget
     /// ([`LazyConfig::max_fetch`]) before they are fetched: an error, and
     /// nothing counted, if they would take it past the budget.
@@ -225,20 +260,70 @@ impl LazyStorage {
     /// Run one pass of `f` over this storage. `Done` when `f` read nothing
     /// that is missing; otherwise `Need` with the ranges to fetch, and `f`'s
     /// result is dropped (it may be an error caused by the miss, or a
-    /// result built around one).
+    /// result built around one). Hints are not followed; see
+    /// [`Operation::attempt`].
     pub fn attempt<T>(&self, f: impl FnOnce() -> T) -> Step<T> {
+        self.attempt_within(f, 0)
+    }
+
+    /// [`attempt`](Self::attempt), adding to a pass that misses the hinted
+    /// blocks it lacks while everything asked for stays within `budget`
+    /// bytes.
+    fn attempt_within<T>(&self, f: impl FnOnce() -> T, budget: u64) -> Step<T> {
         {
             let mut st = lock(&self.state);
             st.missing.clear();
+            st.hinted.clear();
             st.stats.passes += 1;
         }
         let out = f();
-        let missing = std::mem::take(&mut lock(&self.state).missing);
+        let (missing, hinted) = {
+            let mut st = lock(&self.state);
+            (
+                std::mem::take(&mut st.missing),
+                std::mem::take(&mut st.hinted),
+            )
+        };
         if missing.is_empty() {
             return Step::Done(out);
         }
         drop(out);
-        Step::Need(self.runs(missing))
+        let need = self.runs(&missing);
+        if hinted.is_empty() || budget == 0 {
+            return Step::Need(need);
+        }
+        // The hinted blocks still missing, in file order, while they fit.
+        let bs = self.config.block_size;
+        let mut total = ranges_len(&need);
+        let mut wanted = missing.clone();
+        {
+            let st = lock(&self.state);
+            for i in hinted {
+                if wanted.contains_key(&i) || st.blocks.contains_key(&i) {
+                    continue;
+                }
+                let n = (self.len - i * bs).min(bs);
+                if total + n > budget {
+                    break;
+                }
+                total += n;
+                wanted.insert(i, true);
+            }
+        }
+        if wanted.len() == missing.len() {
+            return Step::Need(need);
+        }
+        let with_hints = self.runs(&wanted);
+        // Filling holes between runs can add blocks: never let the hints
+        // take the pass past the budget.
+        if ranges_len(&with_hints) > budget {
+            return Step::Need(need);
+        }
+        {
+            let mut st = lock(&self.state);
+            st.stats.hinted_blocks += (wanted.len() - missing.len()) as u64;
+        }
+        Step::Need(with_hints)
     }
 
     /// The bytes of the file at `offset`, fetched for a range a pass asked
@@ -309,7 +394,7 @@ impl LazyStorage {
     ) -> Result<T, String> {
         let op = self.operation();
         loop {
-            match self.attempt(&mut f) {
+            match op.attempt(&mut f) {
                 Step::Done(v) => return Ok(v),
                 Step::Need(ranges) => {
                     op.charge(&ranges)?;
@@ -350,14 +435,14 @@ impl LazyStorage {
     /// blocks, a one-block hole between two runs filled so they merge
     /// (unless the hole is cached: it would be fetched again), each at
     /// most `max_request` long.
-    fn runs(&self, missing: HashMap<u64, bool>) -> Vec<Range<u64>> {
+    fn runs(&self, missing: &HashMap<u64, bool>) -> Vec<Range<u64>> {
         let bs = self.config.block_size;
         let mut wanted: Vec<u64> = missing.keys().copied().collect();
         wanted.sort_unstable();
         let mut st = lock(&self.state);
         // Remember which blocks only bulk reads asked for: they are kept
         // as bulk once supplied.
-        for (&i, &metadata) in &missing {
+        for (&i, &metadata) in missing {
             if metadata {
                 st.bulk_pending.remove(&i);
             } else {
@@ -499,6 +584,25 @@ impl Storage for LazyStorage {
         self.len
     }
 
+    fn hint(&self, offset: u64, len: usize) {
+        // A hint is about a structure, not bulk data: at most a block or
+        // 1 MiB of it is followed, and at most `MAX_HINTED` blocks a pass,
+        // whatever a hostile file makes a parser hint.
+        let len = (len as u64).min(self.config.block_size.max(1 << 20));
+        let Some(span) = self.span(offset, len) else {
+            return;
+        };
+        let mut st = lock(&self.state);
+        for i in span {
+            if st.hinted.len() >= MAX_HINTED {
+                return;
+            }
+            if !st.blocks.contains_key(&i) {
+                st.hinted.insert(i);
+            }
+        }
+    }
+
     fn read_ranges(&self, ranges: &[Range<u64>]) -> Result<Vec<Cow<'_, [u8]>>, FormatError> {
         let mut spans = Vec::with_capacity(ranges.len());
         let mut total = 0u64;
@@ -797,6 +901,89 @@ mod tests {
         }
     }
 
+    #[test]
+    fn hinted_blocks_come_with_a_miss_and_never_alone() {
+        let data = file(16 * 1024);
+        let s = LazyStorage::new(data.len() as u64, config(1024, 1 << 20));
+        serve(&s, &data, &[3072..4096]);
+        let op = s.operation();
+        // Hints alone: the pass is done, nothing is fetched.
+        let step = op.attempt(|| {
+            s.hint(8 * 1024, 100);
+            owned(s.read_at(3072, 8))
+        });
+        assert!(matches!(step, Step::Done(Ok(_))), "{step:?}");
+        // With a miss, the hinted blocks not cached come too (block 3 is
+        // cached; 10..=11 is one run, 14 another).
+        let pass = || {
+            s.hint(3072, 10);
+            s.hint(10 * 1024 + 1000, 100);
+            s.hint(14 * 1024, 1);
+            owned(s.read_at(0, 8))
+        };
+        let Step::Need(need) = op.attempt(pass) else {
+            panic!("block 0 is missing");
+        };
+        assert_eq!(
+            need,
+            vec![0..1024, 10 * 1024..12 * 1024, 14 * 1024..15 * 1024]
+        );
+        // A plain attempt does not follow hints.
+        let Step::Need(plain) = s.attempt(pass) else {
+            panic!("block 0 is missing");
+        };
+        assert_eq!(plain, vec![0..1024]);
+        serve(&s, &data, &need);
+        let Step::Done(got) = op.attempt(pass) else {
+            panic!("everything was supplied");
+        };
+        assert_eq!(got.unwrap(), &data[..8]);
+        assert_eq!(s.stats().hinted_blocks, 3);
+    }
+
+    #[test]
+    fn hints_stay_within_the_fetch_budget() {
+        // A budget of 3 blocks: the missed block and the first two hinted
+        // ones fit, the rest are left out; a hint never makes a call fail.
+        // (Hinted blocks two apart would be merged with the hole between
+        // them, which would not fit: then no hint is followed.)
+        let data = file(64 * 1024);
+        let mut c = config(1024, 1 << 20);
+        c.max_fetch = 3 * 1024;
+        let s = LazyStorage::new(data.len() as u64, c);
+        let got = s
+            .run_blocking(
+                || {
+                    for i in 0..20 {
+                        s.hint(20 * 1024 + i * 3072, 1);
+                    }
+                    owned(s.read_at(0, 8))
+                },
+                |r| Ok(data[r.start as usize..r.end as usize].to_vec()),
+            )
+            .unwrap()
+            .unwrap();
+        assert_eq!(got, &data[..8]);
+        let st = s.stats();
+        assert_eq!(
+            (st.bytes_fetched, st.hinted_blocks),
+            (3 * 1024, 2),
+            "{st:?}"
+        );
+        // A hint longer than the file, or past its end, is harmless.
+        let s = LazyStorage::new(data.len() as u64, config(1024, 1 << 20));
+        s.run_blocking(
+            || {
+                s.hint(0, usize::MAX);
+                s.hint(u64::MAX, 10);
+                owned(s.read_at(0, 8))
+            },
+            |r| Ok(data[r.start as usize..r.end as usize].to_vec()),
+        )
+        .unwrap()
+        .unwrap();
+    }
+
     #[test]
     fn a_failed_fetch_is_an_error_not_data() {
         let data = file(4096);
diff --git a/crates/clawhdf5-wasm/src/lib.rs b/crates/clawhdf5-wasm/src/lib.rs
index 6a512a2..85271d2 100644
--- a/crates/clawhdf5-wasm/src/lib.rs
+++ b/crates/clawhdf5-wasm/src/lib.rs
@@ -391,7 +391,7 @@ impl Http {
     ) -> Result<T, JsError> {
         let op = storage.operation();
         loop {
-            match storage.attempt(&mut f) {
+            match op.attempt(&mut f) {
                 Step::Done(v) => return Ok(v),
                 Step::Need(ranges) => {
                     op.charge(&ranges).map_err(js_err)?;
diff --git a/crates/clawhdf5/src/reader.rs b/crates/clawhdf5/src/reader.rs
index 4be5f93..7e2ceb5 100644
--- a/crates/clawhdf5/src/reader.rs
+++ b/crates/clawhdf5/src/reader.rs
@@ -372,6 +372,21 @@ impl Storage for FileData {
     fn as_contiguous(&self) -> Option<&[u8]> {
         self.contiguous()
     }
+
+    fn hint(&self, offset: u64, len: usize) {
+        if self.contiguous().is_some() {
+            return;
+        }
+        if let Backing::Storage(s) = &self.backing {
+            // Within the HDF5 data, as a read would be clamped.
+            let size = Storage::len(self);
+            let start = offset.min(size);
+            let len = usize::try_from(size - start).map_or(len, |avail| avail.min(len));
+            if len > 0 {
+                s.hint(self.base + start, len);
+            }
+        }
+    }
 }
 
 /// `bytes`, a backend's answer to a read of `len` bytes, without anything