The primordial curvature perturbation ζ may be generated by some curvaton field σ, which is negligible during inflation and has more or less negligible interactions until it decays. In the current scenario, the curvaton starts to oscillate while its energy density ρ_σ is negligible. We explore the opposite scenario, in which ρ_σ drives a few e-folds of inflation before the oscillation begins. In this scenario for generating ζ it is exceptionally easy to solve the η problem; one just has to make the curvaton a string axion, with anomaly-mediated susy breaking which may soon be tested at the LHC. The observed spectral index n can be obtained with a potential V∝ ϕ^p for the first inflation; p=1 or 2 is allowed by the current uncertainty in n but the improvement in accuracy promised by Planck may rule out p=1. The predictions include (i) running n'≃ 0.0026 (0.0013) for p=1 (2) that will probably be observed, (ii) non-gaussianity parameter f_NL ∼ -1 that may be observed, (iii) tensor fraction r is probably too small to ever observed. ; Comment: 5 pages, 2 figures, revtex, added important results and comments