diff --git a/.github/workflows/run_tests.yml b/.github/workflows/run_tests.yml
index 0137d757..5268212b 100644
--- a/.github/workflows/run_tests.yml
+++ b/.github/workflows/run_tests.yml
@@ -7,7 +7,7 @@ jobs:
   run-tests:
     strategy:
       matrix:
-        python-version: [ "3.10", "3.12" ]
+        python-version: [ "3.11", "3.13" ]
         os: [ ubuntu-latest ]
       fail-fast: false
     runs-on: ${{ matrix.os }}
diff --git a/benchmarks/against_scan.py b/benchmarks/against_scan.py
index 6c67655f..a7771c4e 100644
--- a/benchmarks/against_scan.py
+++ b/benchmarks/against_scan.py
@@ -36,7 +36,7 @@ def speedtest(fn, name):
 # INTEGRATE WITH scan
 
 
-@jax.checkpoint  # pyright: ignore
+@jax.checkpoint
 def body(carry, t):
     u, v, dt = carry
     u = u + du(t, v, None) * dt
diff --git a/diffrax/_step_size_controller/clip.py b/diffrax/_step_size_controller/clip.py
index 0a642d6e..cac6ec1a 100644
--- a/diffrax/_step_size_controller/clip.py
+++ b/diffrax/_step_size_controller/clip.py
@@ -356,7 +356,7 @@ def callback(_keep_step, _t1):
             step_info = None
         else:
             step_index, step_ts = controller_state.step_info
-            # We actaully bump `next_t0` past any `step_ts` whilst checking where to
+            # We actually bump `next_t0` past any `step_ts` whilst checking where to
             # clip `next_t1`. This is in case we have a set up like the following:
             # ```python
             # ClipStepSizeController(
@@ -376,6 +376,24 @@ 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)
+            # 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?
+            #
+            # What's going on here is that we may have
+            # the `diffeqsolve(t0=...)` be prevbefore a jump time (for example due to a
+            # previous diffeqsolve targeting that time), in which case during `.init`
+            # we will obtain `t0 = t1 = prevbefore(jump_time)`.
+            # The `_bump_next_t0` will then move `next_t0` to after the `jump_time`...
+            # whilst leaving `next_t1` unchanged! We actually end up `next_t1 < next_t0`
+            # which is very not okay.
+            #
+            # The fix is to ensure that `next_t1` is itself bumped to at least this
+            # value. As a final detail, we need to make it `nextafter` so that we don't
+            # have a zero-length interval – in this case an underlying PID controller
+            # would just never change the interval size at all, since it acts
+            # multiplicatively. (And even just 1 ULP is enough to unstick it.)
+            next_t1 = jnp.maximum(eqxi.nextafter(next_t0), next_t1)
             made_jump = made_jump | made_jump2
             jump_index = _find_idx_with_hint(next_t0, jump_ts, jump_index)
             next_t1 = _clip_t(next_t1, jump_index, jump_ts, True)
diff --git a/pyproject.toml b/pyproject.toml
index c7ec6d27..459deb3c 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -25,7 +25,7 @@ keywords = ["jax", "dynamical-systems", "differential-equations", "deep-learning
 license = {file = "LICENSE"}
 name = "diffrax"
 readme = "README.md"
-requires-python = ">=3.10"
+requires-python = ">=3.11"
 urls = {repository = "https://github.com/patrick-kidger/diffrax"}
 version = "0.7.0"
 
diff --git a/test/test_adaptive_stepsize_controller.py b/test/test_adaptive_stepsize_controller.py
index 21c24e4a..d785d16e 100644
--- a/test/test_adaptive_stepsize_controller.py
+++ b/test/test_adaptive_stepsize_controller.py
@@ -7,6 +7,7 @@
 import jax.numpy as jnp
 import jax.random as jr
 import jax.tree_util as jtu
+import optimistix as optx
 import pytest
 from diffrax._step_size_controller.clip import _find_idx_with_hint
 from jaxtyping import Array
@@ -361,3 +362,29 @@ def test_jump_at_t1_with_large_t1_in_float32():
         saveat=saveat,
     )
     assert sol.ts == jnp.array([t1])
+
+
+# https://github.com/patrick-kidger/diffrax/issues/713
+def test_t0_at_jump_time():
+    jump_time = 0.98
+    controller = diffrax.PIDController(rtol=1e-6, atol=1e-6)
+    controller = diffrax.ClipStepSizeController(controller, jump_ts=[jump_time])
+    sol = diffrax.diffeqsolve(
+        diffrax.ODETerm(lambda t, y, args: jnp.zeros_like(y)),
+        diffrax.Heun(),
+        t0=eqxi.prevbefore(jnp.asarray(jump_time)),
+        t1=1.2,
+        dt0=None,
+        y0=jnp.array([0, 0, 0, 0.0]),
+        stepsize_controller=controller,
+        event=diffrax.Event(
+            cond_fn=lambda t, y, args, **kw: jump_time - t,
+            root_finder=optx.Newton(atol=1e-4, rtol=1e-4),
+            direction=True,
+        ),
+        max_steps=100,
+    )
+    # And in particular not an event.
+    # What used to happen was something very weird where we'd oscillate across the
+    # jump time.
+    assert sol.result == diffrax.RESULTS.successful