We investigate an extension of the MSSM containing a SU(2) Higgs triplet of zero hypercharge and a gauge singlet. We show that in this model the allowed data for the Higgs boson interaction eigenstates tend to group into separate blocks for SU(2) triplet, doublet and singlet. The triplet sector has two degenerate paired states, each pair composed of a mostly-triplet charged Higgs and a mostly-triplet scalar or pseudoscalar state. The mostly-doublet sector involves instead a SM-like Higgs of 125 GeV and extra mass-degenerate states, composed of a charged, a scalar and a pseudoscalar state. We analyze the exotic decay of the SM-like Higgs into two light (hidden) pseudoscalars, as well as decays into final states containing taus, muons and bottom quarks, and explore the parameter space at the LHC. We also investigate the phenomenology of triplet-Like charged Higgs-boson pairs, decaying into multilepton final states, in different phases of energy and luminosity of the LHC.