I have hitchhiked through Montana many times. It is a good place to hitchhike. The only place over the years that I have noticed panhandlers was in Missoula. The Bitterroot Valley (Missoula to Hamilton to Darby) is probably the warmest valley in Montana. Missoula may be warmer than Billings.
Bozeman has a reputation as being unfriendly to hitchhikers and transients. They tried to start a shelter there a year or so ago, but it closed down because very few people donated to it (this is what I have heard). I have hitchhiked through Bozeman many, many times. Beautiful public library, beautiful scenery. From what I have heard is that the people of Bozeman don't want their town to turn into another Missoula. There are probably a lot more transients in Missoula than in other places in Montana. Billings might have quite a few transients.
Butte gets really cold in the winter, so I don't know why anyone would want to be homeless in Butte in the winter. There is a small mission in Butte; I have stayed there on two occasions.
Helena is a great place; really nice library (I tend to judge a town by its library). They have a really nice Christian mission downtown--God's Love Mission. There is a VA Hospital in Helena, that is why you will see guys going to God's Love Mission for lunch and dinner and then they leave--probably because they are camping out in the woods someplace. I have noticed that with a lot of ex-military homeless: they like to camp out and be away from people and be left alone; they are pretty tough old guys.
If you need a place to sleep on the east side of Billings east of the river. Keep walking east on the railroad tracks (just north of I-90) and you will see a junked camper maybe fifty yards north of the tracks. I have slept there a few times. You are out of the wind, the rain and snow. I tried to clean it up the best that I could. You have to jump over a fence and walk through a small pasture. It is a nice cozy place out of the elements. I have seen lots of hitchhikers in Billings. They have a mission just south of the railroad tracks in downtown Billings--Montana Rescue Mission.