  We present the resummed predictions for inclusive cross section for the
production of Higgs boson at next-to-next-to leading logarithmic ($\rm
\overline {NNLL}$) accuracy taking into account both soft-virtual ($\rm SV$)
and next-to SV ($\rm NSV$) threshold logarithms. We derive the $N$-dependent
coefficients and the $N$-independent constants in Mellin-$N$ space for our
study. Using the minimal prescription we perform the inverse Mellin
transformation and match it with the corresponding fixed order results. We
report in detail the numerical impact of $N$-independent part of resummed
result and explore the ambiguity involved in exponentiating them. By studying
the K factors at different logarithmic accuracy, we find that the perturbative
expansion shows better convergence improving the reliability of the prediction
at $\rm NNLO + \overline{NNLL}$ accuracy. For instance, the cross-section at
$\rm NNLO + \overline{NNLL}$ accuracy reduces by $3.15\%$ as compared to the
$\rm NNLO$ result for the central scale $\mu_R = \mu_F = m_H/2$ at 13 TeV LHC.
We also observe that the resummed $\rm SV + NSV$ result improves the
renormalisation scale uncertainty at every order in perturbation theory. The
uncertainty from the renormalisation scale $\mu_R$ ranges between $(+8.85\%
,-10.12\%)$ at $\rm NNLO$ whereas it goes down to $(+6.54\% , - 8.32\%)$ at
$\rm NNLO + \overline{NNLL}$ accuracy. However, the factorisation scale
uncertainty is worsened by the inclusion of these NSV logarithms hinting the
importance of resummation beyond $\rm NSV$ terms. We also present our
predictions for $\rm SV + NSV$ resummed result at different collider energies.
