We address the problem of estimation extreme quantiles of Weibull tail-distributions. Since such quantiles are asymptotically larger than the sample maxima, 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 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 refined estimator is established, and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to three estimators of extreme quantiles dedicated to Weibull tail-distributions in a simulation study.