Previous post-processing studies on rainfall forecasts using numerical weather prediction (NWP) mainly focus on statistics-based aspects, while learning-based aspects are rarely investigated. Although some manually-designed models are proposed to raise accuracy, they are customized networks, which need to be repeatedly tried and verified, at a huge cost in time and labor. Therefore, a self-supervised neural architecture search (NAS) method without significant manual efforts called AdaNAS is proposed in this study to perform rainfall forecast post-processing and predict rainfall with high accuracy. In addition, we design a rainfall-aware search space to significantly improve forecasts for high-rainfall areas. Furthermore, we propose a rainfall-level regularization function to eliminate the effect of noise data during the training. Validation experiments have been performed under the cases of \emph{None}, \emph{Light}, \emph{Moderate}, \emph{Heavy} and \emph{Violent} on a large-scale precipitation benchmark named TIGGE. Finally, the average mean-absolute error (MAE) and average root-mean-square error (RMSE) of the proposed AdaNAS model are 0.98 and 2.04 mm/day, respectively. Additionally, the proposed AdaNAS model is compared with other neural architecture search methods and previous studies. Compared results reveal the satisfactory performance and superiority of the proposed AdaNAS model in terms of precipitation amount prediction and intensity classification. Concretely, the proposed AdaNAS model outperformed previous best-performing manual methods with MAE and RMSE improving by 80.5\% and 80.3\%, respectively.