Modeling univariate block maxima by the generalized extreme value distribution constitutes one of the most widely applied approaches in extreme value statistics. It has recently been found that, for an underlying stationary time series, respective estimators may be improved by calculating block maxima in an overlapping way. A proof of concept is provided that the latter finding also holds in situations that involve certain piecewise stationarities. A weak convergence result for an empirical process of central interest is provided, and, as a case-in-point, further details are worked out explicitly for the probability weighted moment estimator. Irrespective of the serial dependence, the estimation variance is shown to be smaller for the new estimator, while the bias was found to be the same or vary comparably little in extensive simulation experiments. The results are illustrated by Monte Carlo simulation experiments and are applied to a common situation involving temperature extremes in a changing climate.