Moshe Kohn: The Mishnah B’Rurah doesn’t say to have a double head covering. What he actually quotes (91:6) is a Zohar that one should keep both his head and eyes covered. The most practical and best way to this (which he explicitly says) is with a Tallis. Obviously it will be very hard to put a Tallis over your head and eyes with a hat on.
Optimus: When did a Kippah and Tzitzis stop being enough?