Optical nanotapers fabricated by tapering optical fibers have attracted considerable interest as an ultimate platform for high-efficiency light-matter interactions. While previously demonstrated applications relied exclusively on the low-loss transmission of only the fundamental mode, the implementation of multimode tapers that adiabatically transmit several modes has remained very challenging, hindering their use in various emerging applications in multimode nonlinear optics and quantum optics. Here, we report the realization of multimode submicron tapers that permit the simultaneous adiabatic transmission of multiple higher-order modes including the LP02 mode, through introducing deep wet-etching of conventional fiber before fiber tapering. Furthermore, as a critical application, we demonstrate fundamental-to-fundamental all-fiber third-harmonic generation with high conversion efficiencies. Our work paves the way for ultrahigh-efficiency multimode nonlinear and quantum optics, facilitating nonclassical light generation in the multimode regime, multimode soliton interactions and photonic quantum gates, and manipulation of the evanescent-field-induced optical trapping potentials of atoms and nanoparticles.