I know this thread is older than the hills, but I did want to point out that cougars can now be hunted year-round and you get 2 tags for 1 fee.
It's more complicated than that. But, hey...NM wouldn't have it any other way, right? The more complicated the better as far as the state game laws are concerned.
You get two tags per license. But, you must call to find out if there are any tags left unfilled within the Cougar Management area before you go hunting. Or go to the website: http://www.wildlife.state.nm.us/recreation/hunting/harvest/documents/2013/Cougar_RIB_2013.pdf
They have limits of cougars to be taken in a given zone. Once that limit has been reached, it is illegal to shoot a cougar there.
In other words, they will sell you a license for a fee, but you may not even be able to hunt them. It's up to the hunter to do his investigation before buying the license.