  We present the resummed predictions for inclusive cross section for Drell-Yan
(DY) production up to next-to-next-to leading logarithmic ($\rm \overline{
NNLL}$) accuracy taking into account both soft virtual (SV) and next-to SV
(NSV) threshold logarithms. We restrict ourselves to resummed contributions
only from quark anti-quark ($q \bar q$) initiated channels. The resummation is
performed in Mellin-$N$ space. We derive the $N$-dependent coefficients and the
$N$-independent constants to desired accuracy for our study. The resummed
results are matched through the minimal prescription procedure with the fixed
order results. We find that the resummation, taking into account the NSV terms,
appreciably increases the cross section while decreasing the sensitivity to
renormalisation scale. \textcolor{black}{We observe that, at 13 TeV LHC
energies, the SV+NSV resummation at $\rm \overline{ NLL} (\rm \overline{NNLL})$
gives about 8\% (2\%) corrections respectively to the NLO (NNLO) results for
the considered $Q$ range: 150-3500 GeV}. In addition, the absence of quark
gluon initiated contributions to NSV part in the resummed terms leaves large
factorisation scale dependence indicating their importance at NSV level. We
also study the numerical impact of $N$-independent constants and explore the
ambiguity involved in exponetiating them. Finally we present our predictions
for the neutral Drell-Yan process at various center of mass of energies.
