We address the estimation of extreme quantiles of Weibull tail-distributions. Since such quantiles are asymptotically larger than the sample maximum, their estimation requires extrapolation methods. In the case of Weibull tail-distributions, classical extreme-value estimators are numerically outperformed by estimators dedicated to this set of light-tailed distributions. The latter estimators of extreme quantiles are based on two key quantities: an order statistic to estimate an intermediate quantile and an estimator of the Weibull tail-coefficient used to extrapolate. 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. This analysis is supported by an asymptotic normality result associated with the refined estimator. A data-driven method is introduced for the practical selection of the intermediate sequences and our approach is compared to three estimators of extreme quantiles dedicated to Weibull tail-distributions on simulated data. An illustration on a real data set of daily wind measures is also provided.