Colorado has far more elk than any other state, and it is one of only a few states offering over-the-counter tags to nonresidents. For that reason, hunter pressure is exceedingly heavy on public land. I know of no quality areas close to access roads. However, Colorado’s preference-point system eventually ensures tags for patient hunters in limited-entry units, where hunting is fantastic. The downside is that it can take half a dozen years or more to accumulate enough points to draw a tag in the better units. My suggestion is to keep on collecting points in Colorado until you have enough to be considered for a limited-entry unit.
In the meantime, western Wyoming is a fine place to try for elk, though nonresidents have to draw a tag in a lottery. Odds of drawing a general tag are about one in three. The Bridger-Teton National Forest is my pick for an elk hunt. This forest covers a huge area, essentially from Yellowstone Park south toward Cokeville. My favorite area is the region on either side of Star Valley south of Jackson in the Greys and Salt river drainages. Wyoming requires nonresidents to hire guides in wilderness areas, but there is no wilderness in most of this region, so you can guide yourself. Access is good in many places, thanks to an extensive trail system. Camping is available along roads through the forest, and most successful hunters will hike at least two or three miles to locate elk. This is steep country, with altitudes of 8,000 feet or more. Many outfitters operate here. To avoid competition, simply get away from the major horse trails and hunt the secluded basins and pockets that don’t have good horse access.