We consider supersymmetric models that include particles beyond the minimal supersymmetric standard model (MSSM) with masses in the TeV range, and that couple significantly to the MSSM Higgs sector. We perform a model-independent analysis of the spectrum and couplings of the MSSM Higgs fields, based on an effective theory of the MSSM degrees of freedom. The tree-level mass of the lightest CP-even state can easily be above the LEP bound of 114 GeV, thus allowing for a relatively light spectrum of superpartners, restricted only by direct searches. The Higgs spectrum and couplings can be significantly modified compared to the MSSM ones, often allowing for interesting new decay modes. We also observe that the gluon fusion production cross section of the SM-like Higgs can be enhanced with respect to both the standard model and the MSSM.