DY, please don’t call Charlie & co. yeshiva haters. Dr. Hall is one of the most civil people I have ever met both online and in real life. Plus, as a opponent of the yeshivish hashkafa, these people are too moderate by far! 😉
Regarding the actual point, in line with the Be’er heteiv who says ‘not the kappel you wear inside but the big hat you wear in the street,’ whenever I actually am wearing a dress hat I keep it on for davening. Usually the “big hat I wear in the street” is my kippa though sometime even I will be found in a fedora (though my current one isn’t black)