  We draw attention to the possibility that inflation (i.e. accelerated
expansion) might continue after the end of slow roll, during a period of fast
oscillations of the inflaton field \phi . This phenomenon takes place when a
mild non-convexity inequality is satisfied by the potential V(\phi). The
presence of such a period of \phi-oscillation-driven inflation can
substantially modify reheating scenarios.
  In some models the effect of these fast oscillations might be imprinted on
the primordial perturbation spectrum at cosmological scales.
