Hospitals: Mountain View, Memorial Medical Center
(the latter is more known for hiring new grads if you fall in that category) I've heard Memorial pays better but I have nothing to verify that. Pay starts around the mid-20's depending on experience, education, etc.
Nursing homes: Sagecrest, Good Samaritan, Village at Northrise, Arbors of Del Rey, Casa Del Sol (there are more but these are the ones I can recall off the top of my head). I believe Sagecrest is owned by a chain involved a lawsuit by the state attorney general.
New Mexico sues nursing home chain on care, staff - The Santa Fe New Mexican: Local News
Rehab Hospital of Southern New Mexico
There are also a couple behavioral
health places. I know Memorial has a home
health division and I imagine Mountain View may have one as well.