In this work, we revisited the priority-first sequential-search decoding algorithm proposed in . By adopting a new metric other than the conventional Fano one, the sequential-search decoding in  guarantees the maximum-likelihood (ML) performance, and hence, was named the maximum-likelihood sequential decoding algorithm (MLSDA). In comparison with the other maximum-likelihood decoders, it was shown in  that the software computational complexity of the MLSDA is in general markedly smaller than that of the Viterbi algorithm. A common problem on sequential-type decoding is that at the signal-to-noise ratio (SNR) below the one corresponding to the cutoff rate, the average decoding complexity per information bit and the required stack size grow rapidly with the information length . This problem somehow prohibits the practical use of sequential-type decoding on convolutional codes with long information sequence at low SNRs. In order to alleviate the problem in the MLSDA, we propose in this work to directly eliminate the top path whose end node is Δ-trellis-level prior to the farthest one among all nodes that have been expanded thus far by the sequential search, which we termed the early elimination. Simulations show that a level threshold Δ around three times of the code constraint length is sufficient to secure a near-ML performance. As a consequence of the small early-elimination threshold required, the proposed early-elimination modification not only can considerably reduce the needed stack size but also makes the average decoding computations per information bit irrelevant to the information length.