DY: Offen the Mishna Berurah himself where he discusses hats, he says “K’Derech Sheholchin B’rechov” MB 91:12. I’ve posted it before:
A bigger issue are those who would remove their Shlumpy hat when talking to someone Chashuv (imagine Gov. Deal) due to it’s misshapen form and dust. Certainly not Omed Lifnei HaMelech.
P.S. Lcheorah, it is much more halachic to wear a tie during davening than a hat for the same reason.