What does it mean when you lose your hat?

Solution 1:

Just like the western cliché where someone shoots a cowbow in the hat and it flies away, this is just a visual candy to add some authenticity and fun to the game. It also means that you were a bit off regarding your headshot and aimed a bit too high and shot the hat only (you should be able to shoot a hat with any gun and even your fists). As far as I'm aware of, drinking whiskey only gives back HP, you can only get your hat back when you respawn.

Solution 2:

After reading some update information from the steam page for FoF updates, it appears that hats flying off actually mean something now.

From the page:

New feature: hat-shot.

As you know, hats fall now when hitting head, so it's completely related to damage. This feature will continue the same, but an additional way to drop hats will be added.

A hat-shot will be feasible by just shooting at enemy's hat, without causing any damage. It just requires a new hitbox that covers the hat area.

And what do you gain by dropping other people hats, may you wonder? Just the pleasure of seeing some poor hat-less bastard? No, it will grant some extra notoriety, so to speak, it allows some extra depth as players may wanna drop enemies hat before giving them a ticket to hell. Please watch this video if you wanna check a movie version of it.

Additionally, a new achievement will be added that consist into dropping a few other player's hats without causing any damage, during same life.