When I lived in New Mexico in the late 70's and early 80's and was an EMT, there was base station in Santa Fe that would occasionaly answer us when we could not get through to the Socorro Hospital. At the time we were assigned Med 5 all the time to talk with this hospital. I was an EMT on the Magdalena Volunteer Fire Department.
I have the 1975 New Mexico plan for EMS communications in front of me and it does not mention the call or the dispatch EMS channels of 9 and 10. It also does not show the EMS network being linked via the state's rather extensive microwave system. That was a million years ago when you consider radio systems.
If EMS personnel now call in on Med 9 and are given an assignment on Med 1-8 to contact a hospital then some sort of central control, more common in other states and areas, is being used. My guess, not having been in New Mexico since I moved from there in October, 1981, is that the base station for control of the network would be in Santa Fe. I would think it would identify as "Santa Fe" as the state police, Fish and Game, and State Forestry dispatchers identified with this call.
This guess and the information I have provided are based on 25 year old observations and could very well be worthless. Maybe someone who lives there now can shed some light.