  We investigate the inclusive hadroproduction of a heavy quarkonium ($J/\psi$
or $\Upsilon$), in association with a light-flavored jet, as a testing ground
for the semi-hard regime of QCD. Our theoretical setup is the hybrid
high-energy and collinear factorization, where the standard collinear approach
is supplemented by the $t$-channel resummation of leading and next-to-leading
energy logarithms \`a la BFKL. We present predictions for rapidity and
azimuthal-angle differential distributions, hunting for stabilizing effects of
the high-energy series under higher-order corrections. Our reaction represents
an additional channel to test the production mechanisms of quarkonia at high
energies and large transverse momenta and to possibly shed light on the
transition region from heavy-quark pair production to single-parton
fragmentation of $J/\psi$ and $\Upsilon$ states.
