We consider the curvaton scenario in the context of supersymmetry (SUSY) with gravity-mediated SUSY breaking. In the case of a large initial curvaton amplitude during inflation and a negative order H2 correction to the mass squared term after inflation, the curvaton will be close to the minimum of its potential at the end of inflation. In this case the curvaton amplitude fluctuations will be damped due to oscillations around the effective minimum of the curvaton potential, requiring a large expansion rate during inflation in order to account for the observed energy density perturbations, in conflict with cosmic microwave background constraints. Here we introduce a new curvaton scenario, the phase-induced curvaton scenario, in which de Sitter fluctuations of the phase of a complex SUSY curvaton field induce an amplitude fluctuation that is unsuppressed even in the presence of a negative order H2 correction and large initial curvaton amplitude. This scenario is closely related to the Affleck-Dine mechanism and a curvaton asymmetry is naturally generated in conjunction with the energy density perturbations. Cosmological energy density perturbations can be explained with an expansion rate H≈1012 GeV during inflation.