Enhancing magneto-optical effects is crucial for reducing the size of key photonic devices based on the non-reciprocal propagation of light and to enable active nanophotonics. Here, we disclose a currently unexplored approach that exploits hybridization with multipolar dark modes in specially designed magnetoplasmonic nanocavities to achieve a large enhancement of the magneto-optically induced modulation of light polarization. The broken geometrical symmetry of the design enables coupling with free-space light and hybridization of the multipolar dark modes of a plasmonic ring nanoresonator with the dipolar localized plasmon resonance of the ferromagnetic disk placed inside the ring. This hybridization results in a low-radiant multipolar Fano resonance that drives a strongly enhanced magneto-optically induced localized plasmon. The large amplification of the magneto-optical response of the nanocavity is the result of the large magneto-optically induced change in light polarization produced by the strongly enhanced radiant magneto-optical dipole, which is achieved by avoiding the simultaneous enhancement of re-emitted light with incident polarization by the multipolar Fano resonance. The partial compensation of the magneto-optically induced polarization change caused by the large re-emission of light with the original polarization is a critical limitation of the magnetoplasmonic designs explored thus far and that is overcome by the approach proposed here.