One-sided type schemes are known to be more appropriate for monitoring a process when the direction of a potential mean shift can be anticipated. Furthermore, if the magnitude of the potential mean shift is unknown, it is desired to design a control chart to perform well over a wide range of shifts instead of only optimizing its performance in monitoring a particular mean shift level. The one-sided adaptive truncated exponentially weighted moving average (ATEWMA) X scheme recommended in this paper is a control chart that combines a Shewhart X scheme and a new one-sided EWMA X scheme together in a smooth way for rapidly detecting the upward (or downward) mean shifts. The basic idea of the recommended one-sided ATEWMA X scheme is to truncate the observations (i.e., the sample means X) first, and then to dynamically weight the past observations according to a suitable function of the current prediction error. This helps to improve the sensitivity of the proposed one-sided ATEWMA X scheme for detecting both small and