Fridge Brilliance: A likely reason to explain why Jonah Hex drinks so much is to dull the pain of infections (that and the difficult life of a Bounty Hunter).
Fridge Logic: If you stop and think about it, how is it he doesn't have health problems with that scarred face? With the sanitation conditions and general medical knowledge of the old west that thing must be getting infected CONSTANTLY.
Fighting them off via sheer badassery?
It's a magic scar. There's a reason his last name is Hex.