In order to provide a balanced protection against a range of shift sizes in high-quality processes, a new one-sided adaptive truncated exponentially weighted moving average (ATEWMA) control chart with known and estimated parameters is developed for monitoring time between-events (TBE) data. A dedicated Markov chain model is established for evaluating the run length properties in known and estimated parameters operating conditions. Furthermore, a two-stage optimal design procedure of the proposed scheme is developed based on the average run length (ARL) criteria. Simulation results show that the one-sided ATEWMA TBE scheme with known parameters is superior to its competitors in detecting both upward and downward shifts. Finally, two real data applications are employed to show the implementation of the recommended scheme in the monitoring of TBE data.