He just needs one with a wider brim - the reason I don’t like the black trilby is because the brim’s too narrow and it elongates his face. There are plenty of other more flattering hats available!
I was having a quick google and I think I found part of Sherlock’s disguise after the fall. He clearly used the traffic cone from Barts when he snuck away.
(This hat is only £2.69 on a fancy dress site. I am sorely tempted to buy it.)