diff --git a/diffrax/_step_size_controller/clip.py b/diffrax/_step_size_controller/clip.py
index 4899da4b..14b52cc4 100644
--- a/diffrax/_step_size_controller/clip.py
+++ b/diffrax/_step_size_controller/clip.py
@@ -98,6 +98,17 @@ def _bump_next_t0(next_t0, ts):
     return next_t0, made_jump1 | made_jump2
 
 
+def _drop_step_tangent(next_t0, next_t1, landed):
+    # After a step that landed on a clip time, the proposed step keeps its value but
+    # not its tangent: see the comment at its use in `adapt_step_size`.
+    # (Written so that the value is exactly `next_t1`, and the tangent that of
+    # `next_t0`.)
+    rigid_t1 = jax.lax.stop_gradient(next_t1) + (
+        next_t0 - jax.lax.stop_gradient(next_t0)
+    )
+    return jnp.where(landed, rigid_t1, next_t1)
+
+
 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
@@ -367,7 +378,8 @@ def callback(_keep_step, _t1):
             # propose a step over the interval [something, prevbefore(x)], then on the
             # next step the inner controller will propose a step over [prevbefore(x), x]
             # which definitely isn't desired!
-            _next_t0, _ = _bump_next_t0(next_t0, step_ts)
+            _next_t0, landed_on_step = _bump_next_t0(next_t0, step_ts)
+            next_t1 = _drop_step_tangent(next_t0, next_t1, landed_on_step)
             step_index = _find_idx_with_hint(_next_t0, step_ts, step_index)
             next_t1 = _clip_t(next_t1, step_index, step_ts, False)
             step_info = step_index, step_ts
@@ -376,6 +388,15 @@ def callback(_keep_step, _t1):
         else:
             jump_index, jump_ts = controller_state.jump_info
             next_t0, made_jump2 = _bump_next_t0(next_t0, jump_ts)
+            # The step we just made was clipped to (just before) this jump time, so it
+            # is often very short, whilst the tangent of its length, d(jump) - d(t0),
+            # is O(1). The inner controller proposes the next step as a multiple of
+            # that length, and keeping its tangent would scale the tangents of all the
+            # following step times by (step size / clipped step size): with several
+            # close traced `jump_ts`, the derivative with respect to them blows up.
+            # Instead, the proposed step moves rigidly with the jump time. (Likewise
+            # after landing on a `step_ts` time, above.)
+            next_t1 = _drop_step_tangent(next_t0, next_t1, made_jump2)
             # This next line is to fix
             # https://github.com/patrick-kidger/diffrax/issues/713
             # TODO: should we add this to the `step_ts` branch as well?
diff --git a/test/test_adaptive_stepsize_controller.py b/test/test_adaptive_stepsize_controller.py
index 4a51336f..5fa2500c 100644
--- a/test/test_adaptive_stepsize_controller.py
+++ b/test/test_adaptive_stepsize_controller.py
@@ -239,6 +239,63 @@ def forcing(s):
     assert tree_allclose(finite_diff, autodiff)
 
 
+# The step clipped to just before a jump is often tiny, whilst the tangent of its
+# length, d(jump) - d(t0), is O(1). This used to be carried into all later steps by
+# `PIDController` (whose `dt = prev_dt * factor` only stopped the gradient of `factor`)
+# and grew jump after jump, so that the gradient with respect to several close jump
+# times was wrong by many orders of magnitude. With `step_ts` too, the steps that land
+# on a `step_ts` time did the same.
+@pytest.mark.parametrize("with_step_ts", [False, True])
+@pytest.mark.parametrize(
+    "adjoint",
+    [
+        diffrax.RecursiveCheckpointAdjoint(),
+        diffrax.DirectAdjoint(),
+        diffrax.ForwardMode(),
+    ],
+)
+def test_grad_wrt_close_jump_ts(adjoint, with_step_ts):
+    def run(shift):
+        # A train of 7 pulses 0.04 apart, whose 14 edges all move with `shift`.
+        length = (1.78 - 6 * 0.04) / 7
+        starts = 0.1 + jnp.arange(7.0) * (length + 0.04) + shift
+        ends = starts + length
+
+        def vector_field(t, y, args):
+            y, _ = y
+            forcing = jnp.sum(jnp.where((t >= starts) & (t < ends), 15.0, 0.0))
+            return -20.0 * y + forcing, y**2
+
+        pid_controller = diffrax.PIDController(rtol=1e-10, atol=1e-10)
+        stepsize_controller = diffrax.ClipStepSizeController(
+            pid_controller,
+            jump_ts=jnp.sort(jnp.concatenate([starts, ends])),
+            step_ts=jnp.linspace(0.0, 2.0, 101) if with_step_ts else None,
+        )
+        sol = diffrax.diffeqsolve(
+            diffrax.ODETerm(vector_field),
+            diffrax.Tsit5(),
+            0.0,
+            2.0,
+            None,
+            (0.0, 0.0),
+            stepsize_controller=stepsize_controller,
+            adjoint=adjoint,
+        )
+        _, integral = cast(Array, sol.ys)
+        (integral,) = integral
+        return integral
+
+    shift = 0.013
+    eps = 1e-6
+    finite_diff = (run(shift + eps) - run(shift - eps)) / (2 * eps)
+    if isinstance(adjoint, diffrax.ForwardMode):
+        _, autodiff = jax.jit(lambda s: jax.jvp(run, (s,), (1.0,)))(shift)
+    else:
+        autodiff = jax.jit(jax.grad(run))(shift)
+    assert tree_allclose(finite_diff, autodiff, rtol=1e-4)
+
+
 def test_pid_meta():
     ts = jnp.array([3, 4], dtype=jnp.float64)
     pid1 = diffrax.PIDController(rtol=1e-4, atol=1e-6)