Third-harmonic generation with metallic or dielectric nanoparticles often suffer from, respectively, small modal volumes and weak near-field enhancements. This study propose and demonstrate that a metallic/dielectric hybrid nanostructure composed of a silver double rectangular nanoring and a silicon square nanoplate can be used to overcome these obstacles for enhanced third-harmonic generation. It is shown that the nonradiative anapole mode of the Si plate can be used as a localized source to excite the dark subradiant octupole mode of the Ag ring, and the mode hybridization leads to the formation of an antibonding and a bonding subradiant collective mode, thereby forming anticrossing double Fano resonances. With the strong coupling between individual particles and the effectively suppressed radiative losses of the Fano resonances, several strong hot spots are generated around the Ag ring due to the excitation of the octupole mode, and electromagnetic fields within the Si plate are also strongly amplified, making it possible to confine more incident energy inside the dielectric nanoparticle. Calculation results reveal that the confined energy inside the Si plate and the Ag ring for the hybrid structures can be about, respectively, more than three times and four orders stronger than that of the corresponding isolated nanoparticles, which makes the designed hybrid nanostructure a promising platform for enhanced third-harmonic generation.