We present the differential predictions for the rapidity distribution of a pair of leptons through the Drell-Yan (DY) process at the LHC taking into account the soft-virtual (SV) as well as next-to-soft-virtual (NSV) resummation effects in QCD perturbation theory to next-to-next-to-leading-order plus next-to-next-to-leading-logarithmic ($\mathrm{NNLO}+\overline{\mathrm{NNLL}}$) accuracy. We perform the resummation in two-dimensional Mellin space using our recent formalism [A. H. Ajjath et al., Phys. Rev. D 103, L111502 (2021)] by limiting ourselves to contributions only from quark-antiquark ($q\overline{q}$) initiated channels. The resummed corrections to the fixed-order results are computed through a matched formula using the minimal prescription procedure. We find that the resummation at next-to-leading-logarithmic (next-to-next-to-leading-logarithmic) level brings about 3.98% (1.24%) corrections. We also observe that the sensitivity to the renormalization scale gets improved substantially by the inclusion of NSV resummed predictions at $\overline{\mathrm{NNLL}}$ accuracy. Further, the lack of quark gluon ($qg$) initiated contributions to the NSV part in the $\overline{\mathrm{NNLL}}$ resummed predictions leaves large factorization scale dependence, indicating their importance at NSV level as we go to higher orders in perturbation theory.