The upper confidence reinforcement learning (UCRL2) strategy introduced in <cit.> is a popular method to perform regret minimization in unknown discrete Markov Decision Processes under the average-reward criterion. Despite its nice and generic theoretical regret guarantees, this strategy and its variants have remained until now mostly theoretical as numerical experiments on simple environments exhibit long burn-in phases before the learning takes place. Motivated by practical efficiency, we present UCRL3, following the lines of UCRL2, but with two key modifications: First, it uses state-of-the-art time-uniform concentration inequalities to compute confidence sets on the reward and transition distributions for each state-action pair. To further tighten exploration, we introduce an adaptive computation of the support of each transition distributions. This enables to revisit the extended value iteration procedure to optimize over distributions with reduced support by disregarding low probability transitions, while still ensuring near-optimism. We demonstrate, through numerical experiments on standard environments, that reducing exploration this way yields a substantial numerical improvement compared to UCRL2 and its variants. On the theoretical side, these key modifications enable to derive a regret bound for UCRL3 improving on UCRL2, that for the first time makes appear a notion of local diameter and effective support, thanks to variance-aware concentration bounds.