There is no requirement to wear a hat of any type to shul, certainly not a fedora which was only invented in the late 19th century (and was originally a women’s hat style, popularized by the legendary licentious apostate actress Sarah Bernhardt). Any headcovering will do. The Chofetz Chaim did not wear a fedora (I heard this personally from a Rosh Yeshiva who had learned in Radin as a child.) Rav Soloveitchik often wore straw hats rather than felt hats.
Wear whatever hat will bring you closer to HaShem!