The QCD coupling αs is determined from a combined analysis of experimental e+e− and e±p jet data confronted to theoretical predictions of the energy evolution of the parton-to-hadron fragmentation functions (FFs) moments –multiplicity, peak, width, skewness– at low fractional hadron momentum z. The impact of approximate next-to-leading order (NLO*) corrections plus next-to-next-to-leading log (NNLL) resummations, compared to previous LO+NLL calculations, is discussed. A global fit of the full set of existing data, amounting to 360 FF moments at collision energies √s ≈ 1–200 GeV, results in αs(mZ2) = 0.1189−0.0014+0.0025 at the Z mass.