We report on the observation of conical third, fifth, seventh, and ninth harmonics that gradually emerge during the supercontinuum generation by filamentation of femtosecond midinfrared pulses in lithium strontium hexafluoroaluminate crystal. We show that the generation of conical odd harmonics is an optical signature of light-driven material reorganization in the form of volume nanogratings at the site irradiated by repetitive femtosecond filaments. The angle-resolved spectral measurements demonstrate remarkably broad spectra of individual odd harmonics, benefiting from a spectrally broadened pump pulse (supercontinuum), and reveal that filament-inscribed nanogratings represent photonic structures that are able to provide ultrabroad phase-matching bandwidths covering the wavelength range from the ultraviolet to the near infrared. We propose a scenario that interprets the generation of conical fifth, seventh, and ninth harmonics as nanograting phase-matched cascaded noncollinear four-wave mixing processes.