Some of the most widely-investigated control charting techniques for autocorrelated data are based on time seriesresiduals. If the mean shift in the autocorrelated process is a sudden step shift, the resulting mean shift in theresiduals is time varying and has been referred to as the fault signature. Traditional residual based charts, such as aShewhart, CUSUM, or EWMA on the residuals, do not make use of the information contained in the dynamics ofthe fault signature. In contrast, methods such as the Cuscore chart or Generalized Likelihood Ratio Test (GLRT) doincorporate this information. In order for the Cuscore chart to fully benefit from this, its detector coefficients shouldcoincide with the fault signature. This is impossible to ensure, however, since the exact form of the fault signaturedepends on the time of occurrence of the mean shift, which is generally not knownapriori. Any mismatchbetween the detector and the fault signature will adversely affect the Cuscore performance. This paper proposesa CUSUM-triggered Cuscore chart to reduce the mismatch between the detector and fault signature. A variationto the CUSUM-triggered Cuscore chart that uses a GLRT to estimate the mean shift time of occurrence is alsodiscussed. It is shown that the triggered Cuscore chart performs better than the standard Cuscore chart and theresidual-based CUSUM chart. Examples are provided to illustrate its use.