Recent observations reveal that spectral breaks at ∼GeV are commonly present in Galactic γ-ray supernova remnants (SNRs) interacting with molecular clouds and that most of them have a spectral (E2dF/dE) ‘platform’ extending from the break to lower energies. In Paper I, we developed an accumulative diffusion model by considering an accumulation of the diffusive protons escaping from the shock front throughout the history of the SNR expansion. In this paper, we improve the model by incorporating the finite volume of molecular clouds, demonstrate the model dependence on particle diffusion parameters and cloud size, and apply it to nine interacting SNRs (W28, W41, W44, W49B, W51C, Cygnus Loop, IC443, CTB 37A and G349.7+0.2). This refined model naturally explains the GeV spectral breaks and, in particular, the ‘platforms’, together with available TeV data. We find that the index of the diffusion coefficient δ is in the range 0.5–0.7, similar to the Galactic averaged value, and the diffusion coefficient for cosmic rays around the SNRs is essentially two orders of magnitude lower than the Galactic average (χ∼ 0.01), which is a good indication for the suppression of cosmic-ray diffusion near SNRs.