We explore how thermal Rossby waves propagate within the gravitationally stratified atmosphere of a low-mass star with an outer convective envelope. Under the conditions of slow, rotationally constrained dynamics, we derive a local dispersion relation for atmospheric waves in a fully compressible stratified fluid. This dispersion relation describes the zonal and radial propagation of acoustic waves and gravito-inertial waves. Thermal Rossby waves are just one class of prograde-propagating gravito-inertial wave that manifests when the buoyancy frequency is small compared to the rotation rate of the star. From this dispersion relation, we identify the radii at which waves naturally reflect and demonstrate how thermal Rossby waves can be trapped radially in a waveguide that permits free propagation in the longitudinal direction. We explore this trapping further by presenting analytic solutions for thermal Rossby waves within an isentropically stratified atmosphere that models a zone of efficient convective heat transport. We find that, within such an atmosphere, waves of short zonal wavelength have a wave cavity that is radially thin and confined within the outer reaches of the convection zone near the star's equator. The same behavior is evinced by the thermal Rossby waves that appear at convective onset in numerical simulations of convection within rotating spheres. Finally, we suggest that stable thermal Rossby waves could exist in the lower portion of the Sun's convection zone, despite that region's unstable stratification. For long wavelengths, the Sun's rotation rate is sufficiently rapid to stabilize convective motions, and the resulting overstable convective modes are identical to thermal Rossby waves.