I would bet the pot on the flop. On the turn, a re-raise might be a good way to get info, but it would pot-commit you probably.
I could see a set of sixes being played like this, only hand I would put villain on that beats you. Maybe he limped 8Ts, 86s??? I wouldn't think so. He also could have a whiffed flush draw, a massive overbet on the river usually reeks of that.
I hate these situations. I wouldn't be able to lay this down, I'm just not good enough.