In real-time systems, correctness depends not only on functionality but also on timeliness. A great number of scheduling theories have been developed for verification of the temporal correctness of jobs (software) in such systems. Among them, the Earliest Deadline first until Zero-Laxity (EDZL) scheduling algorithm has received growing attention thanks to its effectiveness in multicore real-time scheduling. However, the true potential of EDZL has not yet been fully exploited in its schedulability analysis as the state-of-the-art EDZL analysis techniques involve considerable pessimism. In this paper, we propose a new EDZL multicore schedulability test. We first introduce an interesting observation that suggests an insight toward pessimism reduction in the schedulability analysis of EDZL. We then incorporate it into a well-known existing Earliest Deadline First (EDF) schedulability test, resulting in a new EDZL schedulability test. We demonstrate that the proposed EDZL test not only has lower time complexity than existing EDZL schedulability tests, but also significantly improves the schedulability of EDZL by up to 36.6 percent compared to the best existing EDZL schedulability tests.