We have performed density functional calculations in conjunction with the linearized Migdal-Eliashberg equations and the functional derivative approach, which takes into account the energy-range dependence of the density of states at the Fermi level (N(ϵ) variable on the scale of phonon energy), to determine the evolution of the critical temperature (Tc) and the isotope effect coefficient (α) of H3S as a function of pressure on its lm3̄m crystal-structure range (from 162 to 250 GPa). Such approach, in comparison with N(ϵ) = N(0) = const., improves the agreement of Tc with available experiments on the whole range of studied pressure. Considering for α two main contributions: one positive coming from the electron-phonon (el-ph) and the other negative from the electron-electron interaction (el-el), we obtained a monotonic decrement as a function of pressure, independent of applied scheme (N(ϵ) or N(0)). However, when N(ϵ) is taken into account, an important renormalization occurs on both contributions, el-ph and el-el, improving the agreement with experimental data, specially for the high-pressure regime. The observed evolution of Tc and α as a function of pressure indicates, thus, the crucial role of the energy-dependence on N(ϵ) for the proper analysis and description of the superconducting state on high-Tc metal hydrides as H3S, by considering the role of its van Hove singularities.