  The energy dependence of the total hadroproduction cross section of
pseudo-scalar quarkonia is computed via matching Next-to-Leading Order (NLO)
Collinear-Factorisation (CF) results with resummed higher-order corrections,
proportional to $\alpha_s^{n}\ln^{n-1}(1/z)$, to the CF hard-scattering
coefficient, where $z=M^2/\hat{s}$ with $M$ and $\hat{s}$ being the quarkonium
mass and the partonic center-of-mass energy squared. The resummation is
performed using High-Energy Factorisation (HEF) in the Doubly-Logarithmic (DL)
approximation, which is a subset of the leading logarithmic $\ln (1/z)$
approximation. Doing so, one remains strictly consistent with the NLO and NNLO
DGLAP evolution of the PDFs. By improving the treatment of the small-$z$
asymptotics of the CF coefficient function, the resummation cures the
unphysical results of the NLO CF calculation. The matching is directly
performed in the $z$-space and, for the first time, by using the Inverse-Error
Weighting (InEW) matching procedure. As a by-product of the calculation, the
NNLO term of the CF hard-scattering coefficient proportional to
$\alpha_s^2\ln(1/z)$ is predicted from HEF.
