Hi, Skraska.
Your Whitney permit is really only good there. If you want to enter/camp in other areas, you will need to get appropriate permits from any one of the various ranger stations along the Eastside (Mono Lake, Mammoth, White Mountain/Bishop).
A big part of this is that each trailhead has their own quotas, and the FS also likes to get an idea of where you are camping, approximately, in case of an emergency.
If you call and speak to my friend John at the FS permit office, he can give you more information.
(760)873-2483.
Have fun!
-L