@Moshe Kohn: The Mishnah B’Rurah (183:11) by Bentsching is quite interesting actually. But he does cite that it is the Minhag to wear an actual hat of some type for Bentsching. That has nothing to do with Shmoneh Esrei though and is a special Din by a Kos Shel B’racha. In fact, to actually follow the Magen Avraham who the M”B quotes, one would have to wear their hat for Kiddush, Havdalah, a Bris, a wedding, the 4 Kosos at the Seder etc. because they are all Kosos Shel Brachah.
@Derech Hamelech: At what other point in history did respectable society worldwide not wear one of those.