Possible, but I'm sure his agent will look for a better situation for next year. Although his chances of being a starter are probably over, some teams may want a transition guy until their rook is ready.
Coaching staff aside, the Saints have one of the most player friendly locker rooms in the NFL. I believe Duncan pointed out that all of the FAs that signed a one-year contract want to come back with the Saints. And as we've seen, all the players genuinely care for each other. And that means a lot. I would argue against most, that Jameis calling the TD in the Atlanta game shows how good the locker room is, not how bad it is - from a player standpoint.
I think that's why he stays unless the Saints decide to move on from his contract. But I don't see that. I think he stays with the Saints for a long time.