Aqueous two-phase extraction has recently been demonstrated as a new method to separate single-wall carbon nanotubes (SWCNTs). In this work, we determined that the mechanism of separation is driven by the hydrophobicity of the surfactant, or combination of surfactants, at the SWCNT surface. This knowledge allowed us to develop a simple approach for obtaining highly enriched single-chirality suspensions in only 1 or 2 steps. These results were obtained by strategically combining multiple surfactants with different diameter-dependent binding affinities for SWCNTs and salts that readjust the surfactant structure within the mixed micelle surrounding the SWCNTs. The procedure is successfully applied to SWCNTs from different sources (CoMoCAT and HiPco) with various diameter distributions (from 0.53 to 1.2 nm). Each separation step is characterized by optical absorption, resonant Raman, and photoluminescence excitation spectroscopies. By determining the SWCNT sorting mechanism, we were able to develop a new set of parameters that separated another chirality.