The goal of this study was to better understand the spatial variations in the daily average, maximum and minimum air temperatures on Jeju Island, South Korea, by interpolating between observed temperature data that are distributed across a mountainous region. This spatial interpolation was performed in two steps: filling in the ungauged station data using principal component regression, and then downscaling the station‐based data to obtain a spatially gridded data set using the Precipitation‐elevation Regressions on Independent Slopes Model. Whereas the conventional practice of interpolating data does not include filling in ungauged data, our approach is particularly useful because most ungauged data correspond to high‐elevation regions. Using the proposed approach, we were able to reasonably construct the spatial distribution of the temperatures on Jeju Island. Our trend analysis showed that the temperatures increased significantly in the summer and decreased in the winter. Furthermore, the positive temperature trends generally become larger at higher elevation when the average positive trends are significant in the summer.