diff --git a/diffrax/_step_size_controller/clip.py b/diffrax/_step_size_controller/clip.py
index 4899da4b..7cc30312 100644
--- a/diffrax/_step_size_controller/clip.py
+++ b/diffrax/_step_size_controller/clip.py
@@ -100,21 +100,14 @@ def _bump_next_t0(next_t0, ts):
 
 def _find_idx_with_hint(t: RealScalarLike, ts: Array | None, hint: IntScalarLike):
     # Find index of first element of `ts` strictly greater than `t`.
-    # Uses a linear search starting from `hint`. The value `hint` is assumed to be in
-    # `{0, 1, ..., len(ts)}`
+    # The value `hint` is assumed to be in `{0, 1, ..., len(ts)}`; it is only used for
+    # its dtype. We use an unrolled binary search rather than a `while_loop` walking
+    # from `hint`: each iteration of a `while_loop` costs a device-to-host
+    # synchronisation on GPU, whilst this is a handful of fused comparisons.
     if ts is None:
         return 0
-
-    def cond_up(_i):
-        return (_i < len(ts)) & (ts[_i] <= t)
-
-    def cond_down(_i):
-        return (_i > 0) & (ts[_i - 1] > t)
-
-    i = hint
-    i = jax.lax.while_loop(cond_up, lambda _i: _i + 1, i)
-    i = jax.lax.while_loop(cond_down, lambda _i: _i - 1, i)
-    return i
+    i = jnp.searchsorted(ts, t, side="right", method="scan_unrolled")
+    return i.astype(jnp.result_type(hint))
 
 
 class ClipStepSizeController(
diff --git a/test/test_adaptive_stepsize_controller.py b/test/test_adaptive_stepsize_controller.py
index 4a51336f..8e643ff9 100644
--- a/test/test_adaptive_stepsize_controller.py
+++ b/test/test_adaptive_stepsize_controller.py
@@ -313,6 +313,21 @@ def test_find_idx_with_hint():
         assert idx == 3  # not 2; we want the first value *strictly* greater.
         idx = _find_idx_with_hint(1.9, ts, hint)
         assert idx == 2
+        assert _find_idx_with_hint(-1.0, ts, hint) == 0
+        assert _find_idx_with_hint(4.0, ts, hint) == 5
+        assert _find_idx_with_hint(7.0, ts, hint) == 5
+
+
+def test_find_idx_with_hint_is_loop_free():
+    # A `while_loop` costs a device-to-host synchronisation per iteration on GPU, and
+    # this search runs on every step of an adaptive solve.
+    ts = jnp.linspace(0.0, 1.0, 100)
+    hint = jnp.array(3)
+    jaxpr = jax.make_jaxpr(lambda t: _find_idx_with_hint(t, ts, hint))(0.5)
+    assert "while" not in str(jaxpr)
+    idx = jax.jit(lambda t: _find_idx_with_hint(t, ts, hint))(0.5)
+    assert idx == 50
+    assert jnp.result_type(idx) == hint.dtype
 
 
 # https://github.com/patrick-kidger/diffrax/issues/607