diff --git a/astropy/coordinates/sky_coordinate.py b/astropy/coordinates/sky_coordinate.py
index ab475f7d0d29..661658b37342 100644
--- a/astropy/coordinates/sky_coordinate.py
+++ b/astropy/coordinates/sky_coordinate.py
@@ -867,10 +867,12 @@ def _is_name(self, string):
         )
 
     def __getattr__(self, attr):
-        """
-        Overrides getattr to return coordinates that this can be transformed
-        to, based on the alias attr in the primary transform graph.
-        """
+        # If this attribute exists on the class (e.g., a subclass property),
+        # try to access it via normal attribute lookup so that any
+        # AttributeError raised by the descriptor propagates unchanged.
+        for cls in type(self).mro():
+            if attr in getattr(cls, "__dict__", {}):
+                return object.__getattribute__(self, attr)
         if "_sky_coord_frame" in self.__dict__:
             if self._is_name(attr):
                 return self  # Should this be a deepcopy of self?
@@ -2220,4 +2222,4 @@ def from_name(cls, name, frame="icrs", parse=False, cache=True):
         if frame in ("icrs", icrs_coord.__class__):
             return icrs_sky_coord
         else:
-            return icrs_sky_coord.transform_to(frame)
+            return icrs_sky_coord.transform_to(frame)
\ No newline at end of file
diff --git a/astropy/coordinates/tests/test_sky_coord_subclass_attr.py b/astropy/coordinates/tests/test_sky_coord_subclass_attr.py
new file mode 100644
index 000000000000..a3196202ec7d
--- /dev/null
+++ b/astropy/coordinates/tests/test_sky_coord_subclass_attr.py
@@ -0,0 +1,19 @@
+# Licensed under a 3-clause BSD style license - see LICENSE.rst
+import pytest
+import astropy.units as u
+from astropy.coordinates import SkyCoord
+
+class CustomCoord(SkyCoord):
+    @property
+    def prop(self):
+        # Access a non-existent attribute to trigger AttributeError from the descriptor
+        return self.random_attr
+
+
+def test_subclass_property_inner_attributeerror_message():
+    c = CustomCoord('00h42m30s', '+41d12m00s', frame='icrs')
+    with pytest.raises(AttributeError) as excinfo:
+        _ = c.prop
+    # Ensure the error message mentions the missing inner attribute, not the property name
+    assert "random_attr" in str(excinfo.value)
+    assert "prop" not in str(excinfo.value)