How do I get Geddy's old hat back?

The second item from the shop after collecting 500 coins is Geddy's old hat. I received the hat, but am no longer wearing it. Is there anyway to put it back on?

It is not on my items screen, and there is no key to toggle the hat in the controls menu.

You can re-equip Geddy's hat by talking to the shopkeeper.

After unlocking the fourth trinket (pirate hat) for 1,000 coins, I was able to re-equip Geddy's hat, despite losing it several hours earlier.

Simply return to the Buccanary's Shop and speak to the shopkeeper. You can then select and re-equip the "Dorky Hat". You can freely swap between Geddy's Hat and the Pirate Hat.

Apparently you've been hit, and this made you lose the hat. According to this post on Reddit:

You'll loose it when you get damaged.

The use of "loose" suggests you won't be able ever to get it back on. Players suggest that if you beat the game with the hat on, meaning you won't get hit at all, something happens.

The answer to your question is: no, you can't equip this hat again (though you can equip another hat if you find one).