As others have mentioned, bear are most likely going to try to avoid you. If you are in a dangerous encounter, your best bet is likely bear spray over a gun. Mountain lions aren't as thick boned and muscled as bear and likely anything that is good to defend against an attack by people will work with them. You are most likely (even in the middle of nowhere) to need to defend against a human attacker if you need a gun in the woods.
That said, a woods gun makes for a good excuse (for me) to carry a revolver. Usually my S&W 625MG, and if I ever put enough rounds through it that I trust it I'll occasionally take my 3" Rossi 461 (if not, I will likely replace all of my snubs with a 3" S&W 686 Plus soon). With the consideration that you are more likely to need it against people, and if you want more capacity, a 10mm with hard-cast bear loads (like the Buffalo Bore Outdoorsman loads) would be acceptable against bear (probably similar to .41mag) and give you semi-auto capacity.
Though, if you want to try to use what you have, I'd go .40S&W (penetration is important against bear) and get the Buffalo Bore Outdoorsman rounds and probably feel as well prepared as with any other handgun.
