Without knowing how Cold you expect it to get when hunting, its difficult to give you a good recommendation. Any oil based lube( mink oil, olive oil, Ballistol,vegetable oils, mineral oil, etc.) should be good down to -20 Degrees Farenheit. There are " Synthentic oils, made for Airplane engine use that can stay "soft" down to -100 Degrees F. and some go further than that! Talk to the suppliers at small airports to find a source of these oils.
Personally, I have experienced outdoor temperatures of -24 degrees F., with windchills of -60 degrees F. That's cold, IMHO! The squeak of rubber boots in snow goes up very high in these conditions.
I had to walk to school, almost a mile away in this condition, but I would not plan to go hunting in that kind of weather, as I expect any game would be bedded down and only moving to find food twice a day, before returning to their beds. Without knowing their exact bedding areas, or having fresh tracks to cut, Hunting would just be great way to see how long it takes to get frostbite, when its that cold!
If you are hunting in conditions where the temperature is zero degrees F., or above, but still below 32 Degrees F. ( zero degrees Centigrade) any patch lube that does NOT use water as part of the formula should work well. I do think Roundball's suggestion to tuck the extra patches into a bag or wrap, and then carry them under your outer coat, so that your own body heat keeps them warm and pliable, is very good advice.
Because I don't like taking off gloves or mittens in that very cold weather, much less touching bare skin to any metal, I recommend you consider using a ball block of wood to hold pre-lubed Patched Round Balls, so that you can simply run the balls out of the block as needed, down the muzzle, with a gloved hand. You can carry the block on a leather thong around your neck, under your coat, or in an inner pocket to keep the lube soft and pliable.
Good hunting. :hatsoff: