Systematic study of near-yrast band structures in odd-mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Pr</mml:mi><mml:mprescripts /><mml:none /><mml:mrow><mml:mn>125</mml:mn><mml:mo>?</mml:mo><mml:mn>137</mml:mn></mml:mrow></mml:mmultiscripts></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Pm</mml:mi><mml:mprescripts /><mml:none /><mml:mrow><mml:mn>127</mml:mn><mml:mo>?</mml:mo><mml:mn>139</mml:mn></mml…

In the present work, basis space in triaxial projected shell-model approach is expanded to include three and five quasiparticle configurations for odd-proton systems. This extension allows us investigate high-spin band structures observed systems up including second band-crossing region and, as a first major application of this development, properties are investigated odd-mass $^{125\ensuremath{-}137}\mathrm{Pr}$ $^{127\ensuremath{-}139}\mathrm{Pm}$ isotopes. It shown that crossings studied isotopes have mixed with crossing dominated by one-proton coupled two-neutron configuration lighter which then changes three-proton increasing neutron number. Furthermore, $\ensuremath{\gamma}$ bands based on states also delineated it predicted these built three-quasiparticle become favored energy heavier region.