I lived the majority of my life in Dillon, growing up and going through school there. Hunted about every inch of beaverhead county and it is some beautiful country!! I echo what JD said about the mule deer. Its sad what the MT FWP has allowed to happen down there. Other than a few big ranches there isn't a mature (yes FWP just because it has 4 points doesn't mean its mature) mule deer buck in the entire county, outside of the Lima Peaks "trophy" area. I had a great friend who drew the LP permit last year and we did see quite a few solid bucks but nothing like the glory years when you'd see 20-25 bucks every day and several trophy bucks.
The elk numbers are still doing ok, and I do still archery hunt in Beaverhead county. I would say that archery hunting is your best bet for a mature bull. When rifle season opens its a rat race with literally thousands of people flying around on 4 wheelers and all the vast majority of elk on are the private ground that doesn't allow hunting within the first couple of days of gun season. There are some places where you might still find elk on public ground but you'll have to work hard and believe me you'll earn a mature bull. They will go as high as possible, even into the 5 foot deep snow, or into the nastiest places imaginable until all the fluorescent orange is gone.
Whitetail numbers are through the roof in the area, but like was stated by guys already they are virtually all on private ground. You can find some checkerboard public stuff or a few BMAs that hold deer but again they'll get pounded by hunters once gun season opens.
Beautiful country, still adequate elk numbers, lots of great local people, and if you're into casting a fly some of the best fly fishing in the entire state flows right through the town of Dillon! j
Maybe I'll see you on the mountain this fall chasing bulls!!