Motivated by applications of large-scale graph clustering, we study random-walk-based LOCAL algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, both in terms of the clustering accuracy and the conductance of the output set. We also prove that our analysis is tight, and perform empirical evaluation to support our theory on both synthetic and real data. More specifically, our method outperforms prior work when the cluster is WELL-CONNECTED. In fact, the better it is well-connected inside, the more significant improvement we can obtain. Our results shed light on why in practice some random-walk-based algorithms perform better than its previous theory, and help guide future research about local clustering.