Yes, definitely try and get an appointment.
I don't imagine it's a sign of any spread - it sounds like it has more to do with the after effects of the treatment and maybe if he feels like crap, then sleeping is a way of alleviating that feeling. However, if he can get a prescription for something to help him feel better then maybe he can start to break the cycle.
Are you able to have a positive conversation with him as to how he is feeling both physically and mentally - I know it's difficult to do without the emotions getting in the way - but try and it will help you prepare for the conversation with the specialists.
Good luck - I know it's a difficult road we tread and sometimes we forget just how hard it is on our partners. x