From Spring, TX to Wyoming.
1st stop: Clayton,NM. Best Western Kokopelli Lodge very pet friendly, even gave Dora a welcome doggie bag with treats, they only charged $10.00 a day for her. After that stop most of the hotels we stayed every night were Super 8, Motel 6, and Holiday Inn. Every hotel is different about their pet policy and they have so many rooms designated for pets or they put you in a smoking room
...the easiest way to work that issue on the road is to use your GPS or phone (web) and call ahead of time if you don't have reservations in advance and ask if they are pet friendly. All state parks are pet friendly and as long as they are keep on the leash they can walk around with the owner, except inside stores and restaurants of course. I spent a lot of years in Germany with the Navy and I remember eating at a restaurant and the guy at the next table had a big dog laying under the table, and that was very common to see all the time, I wish we could get to that point here in the states. I'm sure there will be less depressed people walking around and less suicides, owning a dog makes your life complete.
Picture#1: Royal Gorge #2:Buffalo Bill Historical Center #3: Yellowstone #4: Devil's Tower