I think getting a good AT like a toyo AT3 or similar is better than a bigger tire.
Not sure if it was mentioned, but for us up here in the north where we do get a fair amount of ice in the winters, a bigger/wider tire does give you a larger contact patch, BUT, lowers the PSI of the of the tires interaction with the surface, essentially making the truck lighter, allowing less traction in slick conditions. This is the same reason guys run 12.50's + in the sand, they promote floatation! Not good on ice or on the highway during a downpour.
My .02 cents, A solid All terrain, maybe a size taller for a little added ground clearance, no added width if possible.