The influence of the misfit strain on the lateral shift (Goos–Hänchen effect) experienced by a near-infrared electromagnetic wave upon reflection from the surface of a bilayer consisting of a magnetic, gyrotropic (i.e., whose permittivity tensor elements depend upon magnetization) yttrium–iron garnet film deposited on a nonmagnetic gadolinium–gallium garnet substrate is investigated theoretically. In the geometry of the transverse magnetooptical Kerr effect, it is shown that the mechanical strain near the geometrical film/substrate interface can induce a significant lateral shift of the beam for incidence angles close to normal incidence, where no shift appears in the absence of strain. Our calculations demonstrate positive as well as negative values of the lateral shift, depending on the incident light polarization and on the film thickness. In contrast with that of the misfit strain, the influence of the magnetization of the gyrotropic film on the lateral shift is more noticeable for a TM- than for a TE-polarized wave.