Further improving PhD thesis vs publishing as a post-doc
You can always revise and extend your old work. Make your dissertation work as good as you can in the available time and then move on. If the problems you work on are sufficiently interesting there is always room to continue. 

Whether you continue the work with your advisor or not may be a question that depends on the expectations of your field. 

Most doctoral students hopefully finish with a notebook full of problems that were interesting but couldn't be completed during the graduate program. The ideas don't die just because you graduate.