Depending on what your mountain bike is and how good of one it is - it could be that if you switch to a road or cyclocross bike, you won't need as small of gears to get up the hills - the road or cyclocross bike won't be nearly as heavy, it'll have a lot more efficient tires on it, and it won't have a front suspension, which does make mountain bikes a bit less efficient as climbers if you can't lock out the suspension. It does take some work to move those knobbly mountain bike tires on road.

Are there any bike shops close to a mountain or a big hill that you could take a road or cyclocross bike on a test ride to try out? Or a bike shop that rents bikes or a friend with a road or cyclocross bike? It's hard to directly compare how much effort it takes to take a mountain bike up a hilll vs. a road or cyclocross bike. So you would get a much better idea of it if you just went and test rode some road or cyclocross bikes up a mountain. Maybe you have a friend who'll let you test ride theirs?

You may want to consider a touring road bike - you can put cyclocross style tires on them. Look at the surly long haul trucker or maybe the surly cross check. I think they're both triples with smaller gearing options in the crankset & rear cassette... But I could be wrong about that on the crosscheck. Definitely the long haul trucker is.

Catrin had recently started a ton of threads while she's looking for her touring bike - so they discuss touring bikes in great detail and probably give options about other touring bikes there are (jamis aurora or jamis aurora elite has come up a lot in threads too). I think one is called testing my surly love, another surly love, and then she's got various threads that she's started on gearing to understand. I don't remember if they're posted in the new rider section - but look for threads started by Catrin in the last 2 weeks or so - because she's got a lot of the same concerns that you do.

You can change a bike to have easier gearing - but you may as well buy something with stock gearing that is closer to what you need right now because it can get expensive to do some of the changes.