You might look at areas 67, 68, 69. They have always had leftover archery tags (sept 1-30, any elk) in the past. Lots of wilderness in those areas where you would have to hire a guide, but lots of Multiple Use National Forest, too. I quit hunting area 67 a few years ago because of the grizzlies, but not everywhere in those areas are overrun with bears. Look at the Union Pass area; the griz haven't been quite as numerous there (that could have changed), and there is lots of NF land that isn't wilderness.
I would say to try for the northern Bighorns, but I can't draw that as a resident, so good luck.