Evidence of geologic activity still occurring in the Rio Grande rift (RGR) includes quaternary faulting, seismicity, and widening at a small rate. We map the crustal thickness and seismic velocity ratio to generate constrained crustal model cross sections that highlight the regional extension of the southern RGR (SRGR). Specifically, we computed receiver functions and receiver function stacks for 147 stations from EarthScope USArray Transportable Array and previously collected data in the southwestern United States, and interpolated the crustal and velocity results using a kriging interpolation scheme. We include a new optimized gravity inversion approach using the receiver function results as a constraint for inverted density profiles. We produce constrained crustal models that characterize the SRGR as showing a shallower Moho (30 km) in the region of the SRGR, which is thicker than previously suggested. The crust appears to be delaminated west of the RGR in the Basin and Range Province and thicker east of the RGR toward the Great Plains with a denser lower crustal layer. We also find that the Delaware basin shows a significant geophysical anomaly, likely due to thick saturated sediments. Given that there is no deep mantle anomaly determined from other studies in this region, we suggest that rifting could be the result of small‐scale convection of the uppermost mantle, and we conclude that the RGR is at a minimum the boundary between the Basin and Range Province and the Great Plains, perhaps reflecting the terminus of flat slab subduction.