The method of fabricating composites composed of a glass microfiber matrix and the conductive nanofibers is proposed. The nanofibers incorporating single-walled carbon nanotubes (SWCNTs) and an organic material are spread over the surface of glass microfibers and in some areas between microfibers are coupled together into a branched network. The contributions of intra-CNT and inter-CNT conduction mechanisms in the composites with different contents of SWCNTs are evaluated from the analysis of their impedance spectra. The crossover between two conduction mechanisms is observed at a content of SWCNTs of about 0.1 wt. %.