In a seminal paper (Moser and Tardos, JACM'10), Moser and Tardos developed a simple and powerful algorithm to find solutions to combinatorial problems in the variable Lov{á}sz Local Lemma (LLL) setting. Kolipaka and Szegedy (STOC'11) proved that the Moser-Tardos algorithm is efficient up to the tight condition of the abstract Lov{á}sz Local Lemma, known as Shearer's bound. A fundamental problem around LLL is whether the efficient region of the Moser-Tardos algorithm can be further extended. In this paper, we give a positive answer to this problem. We show that the efficient region of the Moser-Tardos algorithm goes beyond the Shearer's bound of the underlying dependency graph, if the graph is not chordal. Otherwise, the dependency graph is chordal, and it has been shown that Shearer's bound exactly characterizes the efficient region for such graphs (Kolipaka and Szegedy, STOC'11; He, Li, Liu, Wang and Xia, FOCS'17). Moreover, we demonstrate that the efficient region can exceed Shearer's bound by a constant by explicitly calculating the gaps on several infinite lattices. The core of our proof is a new criterion on the efficiency of the Moser-Tardos algorithm which takes the intersection between dependent events into consideration. Our criterion is strictly better than Shearer's bound whenever the intersection exists between dependent events. Meanwhile, if any two dependent events are mutually exclusive, our criterion becomes the Shearer's bound, which is known to be tight in this situation for the Moser-Tardos algorithm (Kolipaka and Szegedy, STOC'11; Guo, Jerrum and Liu, JACM'19).