What fuel efficient cars are best for mountain roads?

I currently have a 2013 Honda Fit Sport, but the front wheel drive puts the car into overdrive everytime we go up an incline in the Smoky Mountains where we live. Even on the highway going on an incline it's hard to stay at the speed limit of 60 miles per hour.
For financing purposes I'd like to keep it in the Honda family also I'd like to keep it as fuel efficient as the Honda Fit. Another concern I have is that going into overdrive is going to cause me to break traction in the winter

It is my opinion that you are using the term "overdrive" incorrectly. Overdrive is a flat surface cruising gear that lowers the RPM's to increase fuel efficiency. What you describe is downshift. The tiny engine of this vehicle is not meant for mountain driving. Unfortunately the Fit was not the wisest choice of vehicles. I find your request rather odd. If you wish for a Honda product, why ask here when you can visit the dealership and speak with a sales expert? As to breaking traction. The function of a tire is to make contact with the ground. If the ground is slippery, the vehicle on top, the brand, the FWD or AWD will not matter much. Most people, and I include myself, drive a FWD vehicle all year round. It take knowledge, training and practice to be aware of the limits related to tire energy. In this photo the driver of the car was going too fast for the traffic and conditions. He did not have the tire energy to stop because of the speed. This happened in dry weather. Luckily, he survived. Go to the Honda dealership and get advise and information and go from there.What fuel efficient cars are best for mountain roads

None my friend. Cars are pretty much designed for flatland not climbing. All econoboxes suffer in hills. Want a machine that gets the same mileage in hills as on flatland? Buy a big block truck

Sounds like you need a Subaru. I have had Hondas, and yes at one point considered myself a "Honda Fanboy", but I have found something a lot better. Subarus are the best, they have Symmetrical All Wheel Drive, and horizontally opposed engines. I gave up a Honda for my Subaru and I'm never going back. The AWD is awesome, it never let's me down. Subarus are more reliable than Hondas, in my opinion. I had to go through multiple transmissions on our Odyssey, but have had literally no trouble with my Subaru Forester. You may want to look into a Subaru Imprezza, or even an XV Cross trek hybrid. Both are the most fuel efficient AWD vehicles on the market right now. Once you buy a Subaru, you won't buy another brand again.

They have 4wd CRVs and Pilots

Hi a toyota landcruiser would be a far better car for the this especially with snow and ice.