Well, you wouldn't want to leave it like that, but it sounds like you are planning to remediate the excess excavation, and like this might be a perfectly suitable foundation for your mild-climate, non-freezing area.
The undermining is minor in scope.
A small heavy hammer (hand sledge, or perhaps a dead blow mallet) might be more effective than "punching it with your hands" for packing back under there, without requiring the room to work that a larger one would.