In the case of Weibull tail distributions, the most commonly used methodology for estimating extreme quantiles is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the Weibull tail coefficient. The common practice is to select the same intermediate sequence for both estimators. We show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined estimator. The asymptotic normality of the latter estimator is established, and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to various bias-reduced estimators in a simulation study.