Why does Jason bother to cover his face at all? He used a potato sack (or whatever it was) in the first couple of films, so it clearly was not for Rule of Cool (not that a maybe undead, psychopathic murderer would/should care about that anyway). His sole purpose in life seems to be to terrorize and/or kill anyone 13 or older that he comes across— his hideous mug would be great for doing the former (as seen in Jason Takes Manhattan). There's no chance he's THAT sensitive about his physical deformities, is there?
Looking for anything resembling genuine characterization in the series is a mostly lost cause, but there are occasional signs throughout the series that at the end of the day, Jason Voorhees is an angry, cruel ten-year-old boy in the body of a seven-foot-tall immortal serial killer. He's actually reasonably sensitive about his disfigurement, to the point where he may wear the mask so he doesn't have to look at himself.
And "Jason Takes Manhattan" has him show he knows how ugly his disfigured face is, considering he lifts his mask to scare away some guys bothering him because he didn't have the time to kill them.