We investigate an extension of the Standard Model with one additional triplet of scalar bosons. Altogether, the model contains four Higgs bosons. We analyze the associated production of the doubly charged scalar with the Standard Model gauge bosons and the remaining Higgs bosons of the model, which are: the light (SM) and heavy neutral scalars and a singly charged scalar. We estimate, in the context of the present (HL–LHC) and future (FCC–hh) hadron colliders, the most promising processes in which a single produced doubly charged Higgs boson is involved.