I've hunted NM 3 times, elk flintlock, mule deer flintlock, and antelope rifle. Also lived on the Ca side near Parker Az for 7 years. Both Az and NM are arid states with elevations from flat low desert to steep very thin air alpine ski areas.
I hunted elk on the Acoma indian reservation and like above there are all those things. You rent a 4wd and follow your guide to the tent camp in the interior of the rez. You have a full size tent/cot. There is a hard trailer with several heated showers/bathrooms and a large dining tent. Camp will probably have 5-6 hunters. Each person has their own tent and section of the Acoma with either a native or non-native guide. You'll go out in his truck, maybe glass vistas, maybe sit near a waterhole, maybe call near an opening. Depends on timing/elk movements/weather.
The Acoma is very well known, they have a high success rate but not 100%. When I was there were 6 hunters and I was the only one to get an animal, the only one with even a shot opportunity on a large bull. But the conditions were unusual. We had severe thunderstorms everyday, even 2 emergency situations. During one of the storms one of the groups didn't come back that evening. Rescue was radioed but the chopper couldn't fly in the dark with the rain, winds, and lightning. They couldn't access their area with 4wd and atvs because of flooding. Luckily the next morning was clear and they were found. Their truck was submerged in a washed out road and they swam to the woods, spent a cold wet night under the trees. The next day one of the native guides had a sandstone cliff give way under him and he fell about 20 feet on rocks, broke his left collar bone and lower arm. Had to be taken over 100 miles to a hospital.
So with water and mud everywhere the elk patterns were unusual. I shot mine during a very intense dry thunderstorm, lightning and thunder like crazy but the rest of the rez was getting a deluge. The following day where I shot mine it would have been inaccessible due to the road now being in the middle of a lake. But that is weather and that is hunting. The Acoma is a stunning place.
People have different tolerances for cost but if you are interested I can connect you with the manager and also a guide there, he's non-native and a friend. Rick Chancellor, also owns Major League Outfitters. I've hunted with him in NM, Texas, and Mexico.
A few pics from the Acoma hunt: