I lived in NM 14 years (Alamogordo) and was a CC Permit Holder for the last two years of my residency.
AFAIK, state law that which supposedly supercede all local laws, a NM Concealed Permit Holder may carry concealed in any establishment that has package sales of Alcohol only, such as Gas Stations, Wally World, Convenience Stores-etc.
No open-concealed carry in Taverns-Bars, or venues that have on site consumption of beer-wine-spirits.
(Such as an example the Alamogordo Sgt Wille Estrada Memorial Civic Center which I worked at for 8 years, and possessed a NM Government Liquor License)
But, management reserves the right to ban such should they choose, and should have signage in place at entrances. Should you be spotted carrying, and asked to leave by management, you likely should with no arguments, otherwise you'll likely be cited for trespassing by LEOs.
It was also my understanding as far as CC rules, that with whatever Pistol(s) you qualified with, with a certified state Instructor, you may carry that caliber, or smaller, but not a larger caliber.
For example, if you qualified with .38-.380 you cannot then conceal carry .40 cal Semi-Auto, or .45 Cal Revolvers.
I qualified with both .45 Auto and .45 Revolver, it stated both Calibers on the back of my Permit, and thus could carry those Calibers, or any smaller Calibers by Concealed Carry.
This rule does not of course apply to Open Carry where permitted, such as Hunting, carry whatever you like and feel you need..