The hadroproduction of the radially excited heavy-quarkonium states ψ(2S) and ϒ(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 αs and the relative heavy-quark velocity v. A satisfactory description of the ψ(2S) transverse-momentum (pT) distributions measured by ATLAS, CMS, and LHCb at center-of-mass energy S=7 TeV is obtained using the color-octet long-distance matrix elements (LDMEs) extracted from CDF data at S=1.96 TeV. The importance of the fragmentation mechanism and the scale evolution of the fragmentation functions in the upper pT range, beyond 30 GeV, is demonstrated. The ϒ(3S) pT distributions measured by CDF at S=1.8 TeV and by LHCb at S=7 TeV and forward rapidities are well described using LDMEs fitted to ATLAS data at S=7 TeV. Comparisons of polarization measurements by CDF and CMS at large pT values with our predictions consolidate the familiar problem in the ψ(2S) case, but yield reasonable agreement in the ϒ(3S) case.