Evolution of the electron spectra with the effective value of the weak axial-vector coupling constant gA was followed for 26 first-, second-, third-, fourth- and fifth-forbidden β− decays of odd-A nuclei by calculating the involved nuclear matrix elements (NMEs) in the framework of the microscopic quasiparticle-phonon model (MQPM). The next-to-leading-order terms were included in the β-decay shape factor of the electron spectra. The spectrum shapes of third- and fourth-forbidden nonunique decays were found to depend strongly on the value of gA, while first- and second-forbidden decays were mostly unaffected by the tuning of gA. The gA-driven evolution of the normalized β spectra was found to be quite universal, largely insensitive to the small changes in the nuclear mean field and the adopted residual many-body Hamiltonian producing the excitation spectra of the MQPM. This makes the comparison of experimental and theoretical electron spectra, coined “the spectrum-shape method” (SSM), a robust tool for extracting information on the effective values of the weak coupling constants. In this exploratory work two new experimentally interesting decays for the SSM treatment were discovered: the ground-state-to-ground-state decays of 99Tc and 87Rb. Comparing the experimental and theoretical spectra of these decays could shed light on the effective values of gA and gV for second- and third-forbidden nonunique decays. The measurable decay transitions of 135Cs and 137Cs, in turn, can be used to test the SSM in different many-body formalisms. The present work can also be considered as a (modest) step towards solving the gA problem of the neutrinoless double beta decay.