We consider the semi-inclusive emission of a Higgs boson in association with a light-flavored jet separated by a large rapidity interval at the LHC. The accessed kinematic regimes fall into the so-called semi-hard sector, whose theoretical description lies at the intersection corner between the collinear factorization and the high-energy resummation. We present a prototype version of a matching procedure aimed at combining next-to-leading fixed-order (NLO) calculations from POWHEG with the resummation of next-to-leading energy logarithms (NLL) as obtained from JETHAD.