We revisit the isocurvature density perturbations induced by quantum fluctuations of the axion field by extending a recently developed analytic method and approximations to a time-dependent scalar potential, which enables us to follow the evolution of the axion until it starts to oscillate. We find that, as the initial misalignment angle approaches the hilltop of the potential, the isocurvature perturbations become significantly enhanced, while the non-Gaussianity parameter increases slowly but surely. As a result, the isocurvature constraint on the inflation scale is tightened as H inf O(100) GeV for the axion decay constant f a 10 10 GeV, near the smaller end of the axion dark matter window. We also derive useful formulae for the power spectrum and non-Gaussianity of the isocurvature perturbations.