Statistical downscaling (SD) derives localized information from larger scale numerical models. Convolutional Neural Networks (CNNs) have learning and generalization abilities that can enhance the downscaling of gridded data (Part I of this study experimented with 2-m temperature). In this research, we adapt a semantic-segmentation CNN, called UNet, to the downscaling of daily precipitation in western North America, from the low resolution (LR) of 0.25-degree to the high resolution (HR) of 4-km grid spacings. We select LR precipitation, HR precipitation climatology and elevation as inputs, train UNet over the subsetted south- and central-western US using Parameter-Elevation Regressions on Independent Slopes Model (PRISM) data from 2015 to 2018, and test it independently in all available domains from 2018 to 2019. We proposed an improved version of UNet, that we call “Nest-UNet”, by adding deep-layer aggregation and nested skip connections. Both the original UNet and Nest-UNet show generalization ability across different regions, and outperform the SD baseline (bias-correction spatial disaggregation) with lower downscaling error and more accurate fine-grained textures. Nest-UNet also shares the highest amount of information with station observations and PRISM, indicating a good ability to reduce the uncertainty of HR downscaling targets.