I would seriously look into Wyoming or a limited 1st season rifle in Colorado. Pick a unit with good amount of public land, that has an average amounts of roads, and some roadless areas. I'm not trying to steer you away from Montana, but I've hunted Colorado quite a lot and Colorado offers a lot of elk in rather small areas. The main problem for non-residents in Montana, which is good and bad, that large areas never see much hunting pressure as compared to Colorado,which gets hunted heavy and either forces the elk to private, or around the public. Montana has large mountain ranges that see concentrated hunting pressure on most every weekend, and smaller herds scattered through heavily timbered areas.
I hunted Montana as a non-resident from 2001 until 2007 (four seasons total) for elk and was never successful. It has taken several years to be able to find elk and have success most years since moving here in 2014. If you have 10 days to 2 weeks to hunt I would Montana and enjoy your hunt. If you want to just kill a bull elk, I'd personally look into Wyoming or Colorado, which offer triple success rates as compared to Montana. The hunting in Montana seems to be directed to resident hunters.