diff --git a/diffrax/__init__.py b/diffrax/__init__.py
index b207173a..6ba2c41e 100644
--- a/diffrax/__init__.py
+++ b/diffrax/__init__.py
@@ -81,4 +81,4 @@
 )
 
 
-__version__ = "0.0.5"
+__version__ = "0.0.6"
diff --git a/diffrax/integrate.py b/diffrax/integrate.py
index 5f9337b8..05d42568 100644
--- a/diffrax/integrate.py
+++ b/diffrax/integrate.py
@@ -88,8 +88,14 @@ def _save(state: _State, t: Scalar) -> _State:
     )
 
 
-def _clip_to_end(tnext, t1):
-    return jnp.where(tnext > t1 - 1e-6, t1, tnext)
+def _clip_to_end(tprev, tnext, t1, keep_step):
+    if tnext.dtype is jnp.dtype("float64"):
+        tol = 1e-10
+    else:
+        tol = 1e-6
+    clip = tnext > t1 - tol
+    tclip = jnp.where(keep_step, t1, tprev + 0.5 * (t1 - tprev))
+    return jnp.where(clip, tclip, tnext)
 
 
 def loop(
@@ -161,8 +167,8 @@ def body_fun(state, inplace):
         # The 1e-6 tolerance means that we don't end up with too-small intervals for
         # dense output, which then gives numerically unstable answers due to floating
         # point errors.
-        tnext = _clip_to_end(tnext, t1)
         tprev = jnp.minimum(tprev, t1)
+        tnext = _clip_to_end(tprev, tnext, t1, keep_step)
 
         # The other parts of the mutable state are kept/not-kept (based on whether the
         # step was accepted) by the stepsize controller. But it doesn't get access to
@@ -403,7 +409,7 @@ def _cond_fun(_state):
 
                     def _body_fun(_state):
                         _step, _t = _state
-                        return _step + 1, _clip_to_end(_t + dt0, t1)
+                        return _step + 1, _clip_to_end(_t, _t + dt0, t1, True)
 
                     compiled_num_steps, _ = lax.while_loop(
                         _cond_fun, _body_fun, (0, t0)
diff --git a/test/test_detest.py b/test/test_detest.py
index 78e40bef..9ca511f1 100644
--- a/test/test_detest.py
+++ b/test/test_detest.py
@@ -404,7 +404,7 @@ def _test(solver_ctr, problems, higher):
                 # build up by t=20.
                 # Teeny-tiny steps fix this.
                 dt0 = 0.000001
-                max_steps = 20_000_000
+                max_steps = 20_000_001
             stepsize_controller = diffrax.ConstantStepSize()
         elif solver_ctr is diffrax.ReversibleHeun and problem is _a1:
             # ReversibleHeun is a bit like LeapfrogMidpoint, and therefore bad over