diff --git a/pytensor/link/jax/dispatch/scalar.py b/pytensor/link/jax/dispatch/scalar.py
index 81f2d6c047..bb63bf3135 100644
--- a/pytensor/link/jax/dispatch/scalar.py
+++ b/pytensor/link/jax/dispatch/scalar.py
@@ -181,7 +181,8 @@ def composite(*args):
 @jax_funcify.register(Second)
 def jax_funcify_Second(op, **kwargs):
     def second(x, y):
-        return jnp.broadcast_to(y, x.shape)
+        _, y = jnp.broadcast_arrays(x, y)
+        return y
 
     return second
 
diff --git a/tests/link/jax/test_scalar.py b/tests/link/jax/test_scalar.py
index 260f198944..8d428f450f 100644
--- a/tests/link/jax/test_scalar.py
+++ b/tests/link/jax/test_scalar.py
@@ -25,6 +25,7 @@
 
 
 jax = pytest.importorskip("jax")
+from pytensor.link.jax.dispatch import jax_funcify
 
 
 def test_second():
@@ -40,6 +41,25 @@ def test_second():
     fgraph = FunctionGraph([a1, b], [out])
     compare_jax_and_py(fgraph, [np.zeros([5], dtype=config.floatX), 5.0])
 
+    a2 = matrix("a2", shape=(1, None), dtype="float64")
+    b2 = matrix("b2", shape=(None, 1), dtype="int32")
+    out = at.second(a2, b2)
+    fgraph = FunctionGraph([a2, b2], [out])
+    compare_jax_and_py(
+        fgraph, [np.zeros((1, 3), dtype="float64"), np.ones((5, 1), dtype="int32")]
+    )
+
+
+def test_second_constant_scalar():
+    b = scalar("b", dtype="int")
+    out = at.second(0.0, b)
+    fgraph = FunctionGraph([b], [out])
+    # Test dispatch directly as useless second is removed during compilation
+    fn = jax_funcify(fgraph)
+    [res] = fn(1)
+    assert res == 1
+    assert res.dtype == out.dtype
+
 
 def test_identity():
     a = scalar("a")