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Stanley, Idaho is a nice place in the Southeast, very near the Sawtooth range and a lot of other beautiful outdoors areas. I've never heard of an LDS influence up around that area. They are mostly down around Idaho Falls and Pocatello.
Cascade and McCall are nice, but they've slowly been going up in price as places to live, and get invaded by tourists and commuters a bit too often for the taste of friends who've lived in those areas. Often, they don't feel as small as they are. Also, they get cold. If you don't mind the cold, you'd probably also enjoy Donnelly. Really, all the places I've recommended get cold in the winter due to elevation.
You might also enjoy the area around Grangeville. I went to the U of Idaho with some folks who grew up there. Nearby are Craigmont and Winschester, which are pretty much farming and blue collar towns. I got to see a lot of those when a friend of my sis's from Craigmont got married. Most of the people around there are Christians, not LDS.
All the areas I've recommended make the LDS unnoticeable even if they are present, have SPECTACULAR outdoors access, and have an obvious quaintness where everyone knows everyone but there are also a diversity of business opportunities for someone like your husband. The one factor is, they'll all have a cold winter... it sort of comes with most of Idaho, and especially the great outdoor access.