  The hadroproduction of the radially excited heavy-quarkonium states
$\psi(2S)$ and $\Upsilon(3S)$ at high energies is studied in the parton
reggeization approach and the factorization formalism of nonrelativistic QCD at
lowest order in the strong-coupling constant $\alpha_s$ and the relative
heavy-quark velocity $v$. A satisfactory description of the $\psi(2S)$
transverse-momentum ($p_T$) distributions measured by ATLAS, CMS, and LHCb at
center-of-mass energy $\sqrt{S}=7$ TeV is obtained using the color-octet
long-distance matrix elements (LDMEs) extracted from CDF data at
$\sqrt{S}=1.96$ TeV. The importance of the fragmentation mechanism and the
scale evolution of the fragmentation functions in the upper $p_T$ range, beyond
30 GeV, is demonstrated. The $\Upsilon(3S)$ $p_T$ distributions measured by CDF
at $\sqrt{S}=1.8$ TeV and by LHCb at $\sqrt{S}=7$ TeV and forward rapidities
are well described using LDMEs fitted to ATLAS data at $\sqrt{S}=7$ TeV.
Comparisons of polarization measurements by CDF and CMS at large $p_T$ values
with our predictions consolidate the familiar problem in the $\psi(2S)$ case,
but yield reasonable agreement in the $\Upsilon(3S)$ case.
