We propose a location-adaptive self-normalization (SN) based test for change points in time series. The SN technique has been extensively used in change-point detection for its capability to avoid direct estimation of nuisance parameters. However, we find that the power of the SN-based test is susceptible to the location of the break and may suffer from a severe power loss, especially when the change occurs at the early or late stage of the sequence. This phenomenon is essentially caused by the unbalance of the data used before and after the change point when one is building a test statistic based on the cumulative sum (CUSUM) process. Hence, we consider leaving out the samples far away from the potential locations of change points and propose an optimal data selection scheme. Based on this scheme, a new SN-based test statistic adaptive to the locations of breaks is established. The new test can significantly improve the power of the existing SN-based tests while maintaining a satisfactory size. It is a unified treatment that can be readily extended to tests for general quantities of interest, such as the median and the model parameters. The derived optimal subsample selection strategy is not specific to the SN-based tests but is applicable to any method that relies on the CUSUM process, which may provide new insights in the area for future research.