Collocations for hat - noun

broad-brimmed, wide-brimmed
floppy
battered
pointed
tall
fur, straw, wool (esp. AmE), woollen/woolen, woolly/wooly (esp. BrE)
baseball, bobble (BrE), bowler, cocked, cowboy, hard, panama, party, peaked, riding, sun, ten-gallon, three-cornered, top, trucker (esp. AmE)
silly, stupid

don, place
He placed a battered felt hat on his head.

doff, remove
raise, tip, touch
The doorman tipped his hat as we entered.

pull down
He pulled his hat down over his face.

be trimmed with sth
The governor wore a cocked hat trimmed with white feathers.

box (usually hatbox)
shop, store (AmE)
stand (usually hatstand)
band, pin (usually hatband, hatpin)