Honestly, a raise and/or extension probably doesn't even matter that much. Hamilton owes his job to Fulmer; Fulmer's extremely close to the boosters who will be the ones who would ultimately have to pressure Hamilton to make a change. Those guys are going to have to turn on Fulmer for a change to be made regardless of what his contract status is.
The only thing a raise/extension does is increase the public pressure on Fulmer in case of a bowl game loss or a bad start to next year. If Hamilton and Fulmer want to defuse the situation, then it sure seems like a weird thing to push through.