A .380 would not be to big to jog with and would stop a dog. Heck I guess some 9’s are as light as some .380s. California prefers you defenseless, though.
I’m a walker; not a jogger. And CCW one of two 5” Kimber .45cal ACPs, w/ 230gr HPs. Had to shoot a rabid raccoon once, but not any dogs, so far.