In this study, we investigate the mass spectrum of π and σ mesons at finite chemical potential using the self-consistent NJL model and the Fierz-transformed interaction Lagrangian. The model introduces an arbitrary parameter α to reflect the weights of the Fierz-transformed interaction channels. We show that when α exceeds a certain threshold value, the chiral phase transition transforms from a first-order one to a smooth crossover, which is evident from the behaviors of the chiral condensates and meson masses. Additionally, at high chemical potential, the smaller the value of α, the higher the masses of the π and σ mesons become. Moreover, the Mott and dissociation chemical potentials both increase with the increase in α. Thus, the meson mass emerges as a valuable experimental observable for determining the value of α and investigating the properties of the chiral phase transition in dense QCD matter. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.