Contact information on CV and cover letter for US mathematics postdoc applicants
As you probably know, your communication with search committees will almost certainly be via email until late in the process. So as long as you use an email address that you will retain access to through the spring, there's not much chance of a communication breakdown.

In fact, you would probably be fine not listing a physical address at all. But since it looks slightly weird not to have a mailing address, go ahead and use your address at your current institution. You will still officially be a student there at the time the applications are submitted, so there's nothing even arguably improper.