The primordial curvature perturbation may be due to a `curvaton' field, which dominates (or almost dominates) the energy density before it decays. In the simplest version of the curvaton model the scale of inflation has to be quite high corresponding to a Hubble parameter H>10^7 GeV. We here explore two modifications of the curvaton model which can instead allow inflation at a low scale. (i) The curvaton is a Pseudo Nambu-Goldstone Boson (PNGB), with a symmetry-breaking phase transition during inflation. (ii) The curvaton mass increases suddenly at some moment after the end of inflation but before the onset of the curvaton oscillations. Both proposals can work but not in a completely natural way. Also, the lower bound on the scale of inflation depends somewhat on the details of the framework used. Nevertheless, we show that inflation with H as low as 1 TeV or lower is possible to be attained.