It is theoretically predicted that an isolated narrow resonance in the elastic scattering of electrons by alkali-metal atoms (M) gives rise to a single maximum in the n dependence of the total cross sections of high-Rydberg atoms A** in the highest allowed angular momentum states with M if the resonance width is roughly equal to or larger than the width of a single peak of the electron momentum distribution of the Rydberg states with highest angular momentum, i.e., I = n -1. This can be used to detect the narrow resonance theoretically predicted in e-M elastic scattering at ultra-low energies.