Once I've got as close as I can to my girl, is there any point spending any more time with her? Or should I just spend time with other girls? Will my girlfriend miss me? I know I can always visit the hot springs with her, but are there any other reasons for us to hang out?
I had to check to make sure Persona 4 followed the same pattern as 3 and 5, which I'm more familiar with, but it appears that it does. That is, you don't get anything for hanging out with your girlfriend (or any social link, for that matter) after they have hit max rank.
no, you don't get anything by spending time with them after maxing SLs.
However, it's worth mentioning that maxed social links will have an additional scene at the end of the game if you talk to them on the final day.