The fine structure of the intensity of very slow electron reflection from the surface of a size-quantized film is considered. It is shown that in the case of grazing incidence the resonance contribution to the fine structure is due to the non-parabolic dispersion of the Rydberg-type bands caused by their hybridization with the states of the size-quantized film. The existence of such bands is supported by calculation of the energy bands lying near the continuous spectrum boundary for a Cu(001) film. It is demonstrated that the scattering intensity of the primary beam may strongly depend on its azimuthal orientation.