The spectral properties of the transverse magneto-optical Kerr effect (TMOKE) in periodic metal–dielectric hybrid structures are studied, in particular with respect to the achievable magnitude. It is shown that the TMOKE is sensitive to the magneto-optical activity of the bismuth-substituted rare-earth iron garnet, which is used as a dielectric material in the investigated structures. For samples with larger Bi substitution level and, consequently, larger gyration constant, the magnitude of the TMOKE increases and reaches 13% in the case of a Bi1.8Lu1.2Fe3.6Al1.4O12 magnetic film. Further, it is demonstrated that the TMOKE vanishes at the high-symmetry points of the Brillouin zone (at the Γ and X points). The main enhancement of the TMOKE takes place near the resonances of the surface plasmon polaritons (SPPs) at the metal/magnetic–dielectric interface. However, near the degenerate resonances of the SPPs at the air/metal and metal/magnetic–dielectric interfaces the TMOKE is increased by the air/metal SPPs as well. This phenomenon is explained in terms of a coupled oscillator model.