Shift it out of overdrive if it has it. Constant shifting between gears is what overheats a transmission. If it keeps shifting to a lower gear put it in that gear manually till the grade eases, then shift it back to a higher gear.

I've been through the Smoky Mountains many a time but I never had my transmission smell like that. I would drive in low gear at times but not go over 35 mph in low. I would change up and drive in D at higher speeds. I would use the brakes off and on when necessary. You just got to figure it out. Good luck!!!

Likely keep it in third of fourth gear. You do NOT want the car to be in an overdrive (lower than 1.00) gear while going up a mountain. Check your owner's manual.

Consider having an aux trans cooler professionally installed

Unless there is something wrong with it you can drive normally. If it smells then there is something wrong with it and you need to get it looked at.

When I go to the Smokies my transmission smells a lot after going up a mountain. What is the best technique to drive so I don't wear out my transmission?

I.e. Low gear, high RPMs, etc?