Usually when I mow a hill for the first time, I walk the area. Actually, I do it on level ground as well. Both areas, I look for rocks, holes, and stuff that could damage a tire.
Some areas, I walk each and every time before hitting it with the tractor. Grass gets high, and then add trees into the mix on a steep slope. I walk in and around the trees looking for widow makers, and downed limbs.
Being in the mountains, I'm sure your also gonna encounter large rocks covered by grass and shrubs as well. One thing I used to do when mowing fields in the mountains of vermont. I would leave the FEL on the tractor. Run it curled back and about a foot off the ground.
The sudden stop when the fel contacted a large rock reminded me where that rock was
I tend to avoid side slope mowing. I usually go up and down a hill. Especially if it's steep. Mild grades I'll consider mowing any direction, but I go by how the tractor feels. And that's after mowing up and down the slope first, so I know there aren't any hidden rocks that I could potentially contact with an up hill tire