Hats are a matter of style, not halacha. The halacha is to dress nicely for Shabbos. What defines Shabbos-dik is dependent on place and time, and is always relative. BTW, that is always true about clothing.
A dark conventional suit, with a fedora, will mark you as someone who wants to be seen as a scholar or even a fanatic if you wear it in a shul where everyone else has no jacket, wears a white shirt without a tie, and a kippah srugah. The very same outfit worn to a shul where the norm is a “long” suit (beckishe, kapote, etc.), and where formal hats (including homburgs, steimels, etc.) are the norm, will indicate that you want to be seen a “modern” person familiar with the ways of the outside world.