Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators.In this presentation, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman 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.This new bias reduction method is fully automatic and does not involve the selection of extra parameters. Our approach is compared to the Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided. This is joint work with Michaël Allouche and Stéphane Girard.