I do have to start by saying that transport from CA to TN is extremely long and may not be possible thru our volunteer pilots who normally fly 200-300 mi. each. It is hard to coordinate the number of pilots needed for that distance to be available at the same time and also to not run into weather issues, mechanical problems and overnight stays for the dog. You may be better off looking into commercial transport.
There is no waiting period. Once he has a health certificate he's good to go.
That said, if you still wish to post a request, I would do it now. It takes time to get all set up with the pilots when there are multiple legs to fill.