Benefiting from large intrinsic nonlinearities, low absorption, and high field enhancement abilities, all-dielectric nanoantennas are considered essential for efficient nonlinear processes at subwavelength volumes. In particular, when the dielectric nanoantenna supports the nonradiating anapole mode, characterized by a minimum in the extinction cross section and a maximum electric energy within the material, third harmonic generation (THG) processes can be greatly enhanced. In this work, we demonstrate that a higher-order anapole mode in a 200 nm thick germanium nanodisk delivers the highest THG efficiency on the nanoscale at optical frequencies. By doubling the diameter of a disk supporting the fundamental anapole mode, we discover the emergence of an anapole mode of higher order, with a valley in the extinction cross section significantly narrower than that of the fundamental anapole. Under this condition, we observe a highly improved electric field confinement effect within the dielectric disk, leading to THG conversion efficiencies as large as 0.001% at a third harmonic wavelength of 550 nm. In addition, by mapping the THG emission across the nanodisk, we are able to unveil the anapole near-field intensity distributions, which show excellent agreement with numerical simulations. Our findings remarkably expand contemporary knowledge on localized modes in dielectric nanosystems, revealing crucial elements for the elaboration of highly efficient frequency upconversion nanodevices.