In Parts I [ Appl. Opt. 58, 6067 (2019)APOPAI0003-693510.1364/AO.58.006067] and II [ Appl. Opt. 61, 10049 (2022)APOPAI0003-693510.1364/AO.474920], we used a coupled optoelectronic model to optimize a thin-film CIGS solar cell with a graded-bandgap photon-absorbing layer, periodically corrugated backreflector, and multilayered antireflection coatings. Bandgap grading of the CIGS photon-absorbing layer was continuous and either linear or nonlinear, in the thickness direction. Periodic corrugation and multilayered antireflection coatings were found to engender slight improvements in the efficiency. In contrast, bandgap grading of the CIGS photon-absorbing layer leads to significant enhancement of efficiency, especially when the grading is continuous and nonlinear. However, practical implementation of continuous nonlinear grading is challenging compared to piecewise-homogeneous grading. Hence, for this study, we investigated piecewise-homogeneous approximations of the optimal linear and nonlinear grading profiles, and found that an equivalent efficiency is achieved using piecewise-homogeneous grading. An efficiency of 30.15% is predicted with a three-layered piecewise-homogeneous CIGS photon-absorbing layer. The results will help experimentalists to implement optimal designs for highly efficient CIGS thin-film solar cells.