Stochastic weather generators are commonly employed to create synthetic sequences of daily weather variables across diverse fields, including hydrological, ecological, and agricultural studies. Realistic precipitation sequences, in particular, serve as essential inputs in numerous modeling frameworks. Generalized linear models (GLMs) that incorporate covariates to capture seasonality and teleconnections represent one effective approach for stochastic weather generation. However, these models often underestimate the interannual variability of seasonally aggregated variables, notably precipitation intensity during wet seasons. Recent methods developed to mitigate the issue of overdispersion have nevertheless struggled to adequately replicate observed precipitation intensities in wet seasons. To overcome this limitation, we propose integrating a modified hybrid gamma and generalized Pareto distribution into the GLM-based weather generator. This enhanced method was evaluated using daily precipitation data from Seoul, Korea, and successfully reproduced realistic precipitation intensities while effectively addressing the overdispersion issue.