Weissman extrapolation device for estimating extreme quantiles 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 work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting Weissman estimator. Our approach is compared to other bias reduced estimators of extreme quantiles both on simulated and real data.