That depends on the province where you are applying as you usually need a residence certificate and sometimes you can only get that on a long term visa
What Ellie says is correct if you are still in Bangkok at the time you apply for the extension. If you move to another province you apply at the Immigration Office for that province, not return to Bangkok.
The re-entry permit operates up to your current entry stamp date, no other date matters. You have to re-enter before your current entry until stamp date to maintain that status.