  We revisit the growth of curvature perturbations in non-minimal curvaton
scenario with a non-trivial field metric $\lambda(\phi)$ where $\phi$ is an
inflaton field, and incorporate the effect from the non-uniform onset of
curvaton's oscillation in terms of an axion-like potential. The field metric
$\lambda(\phi)$ plays a central role in the enhancement of curvaton field
perturbation $\delta\chi$, serving as an effective friction term which can be
either positive or negative, depending on the first derivative
$\lambda_{,\phi}$.Our analysis reveals that $\delta\chi$ undergoes the
superhorizon growth when the condition $\eta_\text{eff} \equiv - 2
\sqrt{2\epsilon} M_\text{Pl} { \lambda_{,\phi} \over \lambda} < -3$ is
satisfied. This is analogous to the mechanism responsible for the amplification
of curvature perturbations in the context of ultra-slow-roll inflation, namely
the growing modes dominate curvature perturbations. As a case study, we examine
the impact of a Gaussian dip in $\lambda(\phi)$ and conduct a thorough
investigation of both the analytical and numerical aspects of the inflationary
dynamics.Our findings indicate that the enhancement of curvaton perturbations
during inflation is not solely determined by the depth of the dip in
$\lambda(\phi)$. Rather, the first derivative $\lambda_{,\phi}$ also plays a
significant role, a feature that has not been previously highlighted in the
literature. Utilizing the $\delta \mathcal{N}$ formalism, we derive analytical
expressions for both the final curvature power spectrum and the non-linear
parameter $f_\text{NL}$ in terms of an axion-like curvaton's potential leading
to the non-uniform curvaton's oscillation. Additionally, the resulting
primordial black hole abundance and scalar-induced gravitational waves are
calculated, which provide observational windows for PBHs.
