Laser-induced fluorescence spectroscopy (LIFS) is actively used for remote sensing of atmospheric aerosols and currently is one of the most sensitive and selective techniques for determining small concentrations of substances inside particles. The use of high-power femtosecond laser sources for LIFS-based remote sensing of aerosol contributes to the development of new-generation fluorescence atmospheric lidars since it makes it possible overcoming the energy threshold for the nonlinear-optical effects of multiphoton absorption in particles and receiving the emission signal at long distances in the atmosphere. Our study is aimed to the development and experimental demonstration the technique of nonlinear laser-induced fluorescence spectroscopy (NLIFS) based on the remote excitation of aerosol fluorescent emission stimulated by a spatially-structured high-power femtosecond laser pulse. Importantly, for the first time to our knowledge, we demonstrate the advances of using stochastically-structured plasma-free intense light channels (postfilaments) specially formed by propagation of femtosecond laser radiation through a turbulent air layer for improving NLIFS efficiency. A multiple increase in the received signal of two-photon-excited fluorescence of polydisperse dyed aqueous aerosol by the structured postfilaments is reported.