In this article we present an updated spectrum-shape analysis of the ¹¹³Cd fourfold forbidden non-unique β-decay transition in order to address the quenching of the weak axial-vector coupling g_A in low-momentum exchange nuclear processes. The experimental data were collected in a dedicated low-threshold run with the COBRA demonstrator at the LNGS and resulted in 44 individual ¹¹³Cd spectra. These data are evaluated in the context of three nuclear model frameworks based on a revised version of the spectrum-shape method and the conserved vector current hypothesis. The novel idea devised in the present work is to fit the value of the small relativistic nuclear matrix element (s-NME) driving the nuclear model calculations, which remained essentially as a free parameter in previous studies. This is done by tuning the nuclear structure calculations and making use of the interplay of g_A and the s-NME such that the experimentally known ¹¹³Cd half-life gets reproducible by the different frameworks. In this way, a best fit s-NME value can be derived for each of the considered nuclear models, which finally enters the template calculations used to perform the spectrum-shape analysis for each of the obtained ¹¹³Cd spectra. The primary analysis strategy results in significantly quenched values of the axial-vector coupling for all three nuclear models: g‾_A(ISM)=0.907±0.064, g‾_A(MQPM)=0.993±0.063 and g‾_A(IBFM-2)=0.828±0.140. Moreover, with our data-driven approach one of the main shortcomings of the spectrum-shape method has been resolved. This achievement is a milestone in the description of strongly forbidden β-decays and adds to the indications for the existence of a quenching of g_A in low-momentum exchange nuclear processes.