Search on "stingy brim" if you're thinking of the narrow brimmed hats felt hats like Sinatra wore.
A lot of people don't know this but any fur felt hat (not wool) can be easily trimmed down to any width. One of those tools they use to cut leather belt strips works, but a steady hand and razor knife works too. Just use 220 grit wet/dry sandpaper to smooth/round off the cut edge.
Keep a lookout for a wide-brimmed Stetson hat at garage sale or thrift shop, trim the brim, steam the crown over a teakettle into the shape you want, and there ya go.
Doug