Ruidoso would be another option for a stay if you're heading to White Sands. I rented a cabin in Ruidoso for a week with my two dogs and made the drive to White Sands taking some of the less traveled back roads. You can head over toward the Inn of the Mountain Gods in Mescalero on US 40 and continue on to US 70, which will take you into Alamogordo or take US 40 to NM 244 before you reach Mescalero and wind through the mountains to Cloudcroft, NM (8,600' elev.) then make your way down to Alamogordo on US 82.
Keep an eye out for a turnoff on the south side of US 82 that provides an overlook of Alamogordo and White Sands NM almost 4,000' below as you're heading down US 82.
As digitaldog mentioned, if you're traveling with pets, La Quinta is a great option as I believe they all allow pets at no extra charge - some other motels charge up to $20.00 - $25.00 extra per night per pet, which can really add up. Since I was in the area for a week, I elected to search for a cabin that allowed pets - there are a number of them around Ruidoso. I chose one just north of town as the ones closer into Ruidoso were in much more congested locations. Ruidoso is also great because at almost 7,000' elevation, temperatures for sleeping at night are very comfortable.
Alamogordo is high desert while the area around Ruidoso is alpine. There are also a number of back roads to the north out of Ruidoso that provide scenic options.
On another week trip I rented a cabin in Red River, NM - elev. 8,750', which is a ski area on the north side of the scenic loop around Wheeler Peak - the highest point in the state and well away from Taos. No offense to anyone, but I personally didn't care for Taos - the traffic in town was brutal (I think I waited through 4 or 5 light changes by the espresso shop) and it reminded me of SoCal - a lot of posing going on there.
Heading south from Red River, there's an impressive Vietnam Veteran's Memorial off US 64 just before (or after depending on your direction of travel) the Angel Fire, NM turnoff on NM 434. This was a surprise when I was exploring the area. The memorial sits up on the side of the hill all by itself with views overlooking the valley and is very impressive and unexpected.
Angel Fire is an extremely nice area - to my mind much nicer though dramatically less developed than Taos. Angel Fire is much more laid back and nowhere near the crowds or commercialism of Taos.
The drive south on NM 434 from Angel Fire to N.M 518 was very scenic. I continued on NM 518 to Las Vegas, NM and found it to be a quaint and growing town, which I wasn't expecting.
There's a dramatic difference in environmental characteristics between the mountains around Ruidoso and Cloudburst. They seem to be much drier than the mountains to the north around Angel Fire and Red River.
