In supersymmetric scenarios with a low scale of SUSY breaking [sqrt{F}=O(TeV)] the conventional MSSM Higgs sector can be substantially modified, mainly because the Higgs potential contains additional effective quartic terms. The Higgs spectrum can be dramatically changed, and the lightest state can be much heavier than in usual SUSY scenarios. Novel opportunities for electroweak breaking arise, and the electroweak scale may be obtained in a less fine-tuned way for wide ranges of tan(beta) and the Higgs mass, offering a possible solution to the MSSM fine-tuning problem.