You need to further define "room" You can almost always find something along the lines of Motel-6. I haven't traveled your route, so not sure of those.
Used to be dad would pile us all into the van, and head either to Oregon via 1, or Iowa via 80. We'd drive until he got tired, then look for an exit sign that indicated hotels. Find a cheap one, sleep, then get up and head out again. We even put quarters in the bed shaker a few times!
As recently as about 2010 we traveled to the Grand Canyon and just came over some rise and found a town spread out before us. That's where we stopped. Wife had tried to call a few hotels that were indicated on the Around Me app, but she isn't very good at map reading and kept calling places that were not along our path of travel.