The current experimental lower bound on the Higgs mass significantly restricts the allowed parameter space in most realistic supersymmetric models, with the consequence that these models exhibit significant fine-tuning. We propose a solution to this `supersymmetric little hierarchy problem'. We consider scenarios where the stop masses are relatively heavy - in the 500 GeV to a TeV range. Radiative stability of the Higgs soft mass against quantum corrections from the top quark Yukawa coupling is achieved by imposing a global SU(3) symmetry on this interaction. This global symmetry is only approximate - it is not respected by the gauge interactions. A subgroup of the global symmetry is gauged by the familiar SU(2) of the standard model. The physical Higgs is significantly lighter than the other scalars because it is the pseudo-Goldstone boson associated with the breaking of this symmetry. Radiative corrections to the Higgs potential naturally lead to the right pattern of gauge and global symmetry breaking. We show that both the gauge and global symmetries can be embedded into a single SU(6) grand unifying group, thereby maintaining the prediction of gauge coupling unification. Among the firm predictions of this class of models are new states with the quantum numbers of 10 and under SU(5) close to the TeV scale. The Higgs mass is expected to be below 130 GeV, just as in the MSSM.