There are a lot of Elk around Springerville, AZ on the Apache Indian reservation. Not sure of any remote strips there you can fly in to, or what licenses are required to hunt them.
It surprises me that there is any good hunting on an a Native American reservation. The standard practice was to provide land that was stripped of all of the resources or veritably useless land. Then I thought, if there is good hunting, it would probably be an insult to the Apache Nation to hunt the land. However, after I googled it I discovered that the Reservation is managed with a recreation and wildlife department that encourages tourism and outdoor activities like hunting fishing and wildlife viewing/watching and that Elk hunting revenues are used to manage the herds and the eco system of the game lands. They also report that elk on the reservation have some of "largest antler bearing elk in the world" according to the site.
Sorry about the digression from the topic.