Here's a theory (albeit not a very likely one):
Miles agrees in principle to a contract with Michigan before the SECCG, when it looks like they have no shot at the NCG. Miles tells LSU's AD, he is pissed but can't do anything about it (if I remember, Les wasn't too popular in BR for a week there anyways). Word leaks out Saturday before the game, and the AD tells Miles he still has an obligation to the school for the SECCG, Les agrees and holds that press conference to deny the reports, which he didn't do a great job of anyways. They play the SECCG, and the LSU administration is ready to bid Les goodbye. But then a funny thing happens- WVU and Missouri both lose, leaving LSU in the drivers seat for the NCG.
Miles has already agreed with Michigan to become the next coach, and thats where he wants to be. But he realizes his shot at a national championship is very real, and he wants to win one, plus you can't leave a team right before the NCG. He talks this through with LSU's AD, and they agree that something definitive has to be done to quell all the Les to AA rumors so that he and his team can concentrate on the game. So Les signs a contract extension that changes absolutely nothing, and on the surface it appears Les is staying at LSU. Michigan interviews other "candidates" who have absolutely no interest in the job (Schiano) or who they would never offer the job to (Hoke) to look like they are doing something, but their coach is coming up on 1/8 anyways.
Clearly this is very unlikely, the biggest reason being that it would have leaked somewhere unless the only people who know about it are Les, Bertman, Martin, and MSC. But I mean hell, it could happen.