  We re-analyze Tavatron data on charmonium hadroproduction in the framework of
the color-octet model implemented in the event generator PYTHIA taking into
account initial-state radiation of gluons and Altarelli-Parisi evolution of
final-state gluons fragmenting into $c\bar{c}$ pairs. We obtain new values for
the color-octet matrix elements relevant to this production process. We discuss
the sensitivity of our results to the transverse momentum lower cut-off
employed in the generation to avoid the problematic $p_t{\to}0$ region, arguing
about the reliability of our previous extraction of the NRQCD matrix elements
for the $^3S_1^{(8)}$ and $^1S_0^{(8)}+^3P_J^{(8)}$ contributions. Finally we
extrapolate to LHC energies to get predictions on the $J/\psi$ direct
production rate.
