the regional pilots that I know or have met, either due to 1) Not enough time or 2) A desire for an outside life, nobody is chasing flying gigs during their days off. If you have a BF/GF/Spouse, even less so.
Remember the liability involved, say on your day off, you are teaching for FREE (thus no compensation and possibly technically in compliance with your employer's policy), BUT, your student busts an altitude or something happens. Now you have a potential FAA inquiry. All while "flying on the side" and apps into the majors for interviews.
Is this worth it ?
Again, I have not seen many regional guys doing this. Major airline guy who is at his retire-from-here place, with ample days off, and salary to match yes, several examples, but mostly recreationally, not for pay.
But, good luck