We demonstrate a concept for the giant enhancement of the transverse magneto-optical Kerr effect (TMOKE) using bulk plasmon polariton (BPP) modes in non-magnetic multilayer hyperbolic metamaterials (HMMs). Since the BPP modes are excited through the attenuated total reflection (ATR) mechanism, using a Si-based prism-coupler, we considered a single dielectric magneto-optical (MO) spacer between the prism and the HMM. The working wavelength was estimated, using the effective medium approach for a semi-infinite dielectric-plasmonic multilayer, considering the region where the system exhibits type II HMM dispersion relations. Analytical results, by means of the scattering matrix method (SMM), were used to explain the physical principle behind our concept. Numerical results for giant TMOKE values (close to their maximum theoretical values, ±1) were obtained using the finite element method (FEM), applying the commercial software COMSOL Multiphysics. Our proposal comprises a simple and experimentally feasible structure that enables the study of MO phenomena in HMMs, which may find application in future nanostructured magnetoplasmonic metamaterials for active nanophotonic devices.