A new approach has been used to explain the experimental data for the C-12 + C-12 system over a wide energy range in the laboratory system from 32.0 MeV to 126.7 MeV. This new coupled-channel-based approach involves replacing the usual first-derivative coupling potential by a new, second-derivative coupling potential. This paper first shows and discusses the limitation of the standard coupled-channel theory in the case where one of the nuclei in the reaction is strongly deformed. Then, this new approach is shown to improve consistently the agreement with the experimental data: the elastic scattering and single-2(+) and mutual-2(+) excitation inelastic scattering data as well as their 90 degrees elastic and inelastic excitation functions with little energy-dependent potentials. This new approach makes major improvement on all the previous coupled-channel calculations for this system.