We present a new proposal for enhancing the transverse magneto-optical Kerr effect (TMOKE) in the transmission mode using dielectric magneto-optical (MO) ribs deposited on a high-refractive-index (HRI) slab. Our concept is based on the phase-matching condition between the MO grating and the slab to produce extraordinary TMOKE amplitudes, which are superior to the most successful plasmonic approaches, and at least one order of magnitude higher than using dielectric MO gratings. It is also significant that there was almost no loss in the structure proposed owing to the low level of losses from the constituent materials. These features can be exploited in sensing and biosensing, as we demonstrated by considering the nanostructure in a typical liquid medium.