R’ Schachter himself wears a hat but if you ask him if you have to he says no. He says when he was growing up you had to for Davening because that was a Kavod’dik way of dress. Nowadays it’s not. He still wears his because he’s used to it and would feel uncomfortable without it. But he says you don’t have to anymore.
Oh, and the Mishnah B’rurah does not say, anywhere, that you need a double covering for Davening. It is a myth and a gross misinterpretation (and I don’t think it’s unintentional on the part of some) of 91:2.