The Weibull tail-coefficient (WTC) is the index of regular variation in a regularly varying cumulative hazard function. Due to the specificity of the WTC, and its deep and explicit link to a positive extreme value index (EVI), any estimator of a positive EVI, like all generalizations of the classical Hill estimator, can be used for the estimation of the WTC. These estimators are scale invariant but not location invariant, contrarily to the location/scale invariance of the EVI and of the WTC parameters. With PORT standing for peaks over random thresholds, new classes of consistent PORT WTC-estimators, dependent on an extra tuning parameter s, \(0 \leq s < 1\) , are introduced. These WTC-estimators are highly flexible and are further studied for finite samples, through a Monte-Carlo simulation study. Possible choices of the tuning parameters under play are put forward, and some concluding remarks are provided.