For debugging effectively the shared-memory programs with nested parallelism, it is important to detect efficiently the first races which incur non-deterministic executions of the programs. Previous on-the-fly technique detects the first races in two passes, and shows inefficiencies both in execution time and memory space because the size of an access history for each shared variable depends on the maximum parallelism of program. This paper proposes a new on-the-fly technique to detect the first races in two passes, which is constant in both the number of event comparisons and the space complexity on each access to shared variable because the size of an access history for each shared variable is a small constant. This technique therefore makes on-the-fly race detection more efficient and practical for debugging shared-memory programs with nested parallelism.