Hi! I've looked into this quite a bit and I don't think this is something that's currently implemented. If i'm wrong, though, please feel free to correct me.
Currently, you pretty much have to "choose" between having an overlay and a CT on a face. Not literally, of course, but overlays and CTs don't interact, meaning that you're stuck these options:

1. Have overlays, but any edges without overlays will be totally bare.
   
2. Proper connected textures on the face, but no overlays.
   
3. Both overlays and connected textures, but implemented in a way that may be very undesirable in use cases like this one.
   

With that said, I think it'd be great to have an option that "combines" CT and overlays in some way. That way, you can still have overlays while having some sort of fallback option for air or other unsupported blocks.

I'm aware that similar functionality will probably be possible in the "connection control" update, since you can just have your CT connect to the same blocks as your overlay does. However, I think a method specifically designed for this opens the door for additional features, such as custom "taper-off" tiles for overlays to transition into perpendicular CT edges:

I'm not sure if this would be overly complicated or even impossible with however CTM and overlay are implemented, but I figured it'd be worth it to ask. Would be a game changer for stylized resource packs where outlines are very important to define edges of blocks.