In an effort to elucidate the rich band structures observed in odd-neutron systems, triaxial projected shell model approach is extended to include three-quasineutron and five-quasiparticle configurations. This extension makes it possible to investigate the high-spin states up to and including the second band crossing. Detailed investigation has been performed for odd-mass Xe isotopes with the extended basis, and it is shown that character of the band crossing along the yrast line changes with shell filling of the 1h 11/2 orbital. Further, it is observed that the three-quasiparticle state that crosses the ground-state configuration, leading to the normal band crossing phenomenon along the yrast line, first crosses the γ band based on the ground-state configuration at an earlier spin value. This crossing feature explains the occurrence of the signature inversion observed in the γ bands for some of the studied isotopes.