"I caught a good one, it looked like it could run Up on its back, and away I did ride Just as fast as I could from The West Texas town of El Paso Out to **the badlands of New Mexico**"
The hills are definitely in Mexico. The “badlands” were probably also in Mexico (not New Mexico), since badlands refers to steep, heavily eroded areas with little vegetation. That kind of terrain doesn’t happen in the immediate area until crossing into Mexico.
In this google maps fragment you can see New Mexico is effectively across the street from Rosa’s and the hills are only a couple miles away in Mexico proper.
Here is the topographical layout:
Here’s the schematic:
Here you see Texas, New Mexico, and Mexico all close together; we call the greater region_ the Borderlands_. We are used to people going back and forth. We get along great with each other and wonder why folks who don’t live near the border get so cranked up about it.