Symmetric block-wise concatenated Bose-Chaudhuri-Hocquenghem (SBC-BCH) codes are known to provide strong error-correcting performance with an iterative hard-decision decoding (IHDD). This work shows that some properties of SBC-BCH codes allow one to readily increase the error-correcting capability of constituent BCH codes by one, i.e., from t to$t + 1$, which greatly improves the error-rate performance of SBC-BCH codes. To this end, we propose a new decoding algorithm that utilizes structural features of SBC-BCH codes in conjunction with an extension of the Berlekamp-Massey (BM) algorithm, namely the one-step-ahead (OSA) BM algorithm. This combination enables us to reduce the computational complexity and combat the inherent decoding ambiguity associated with the OSA-BM algorithm. Furthermore, we develop an analytical framework to evaluate the average number of candidate codewords returned by the OSA-BM algorithm and derive an upper bound on the probability of ambiguity. Then, we propose a decoding algorithm, called OSA-aided IHDD and conduct extensive performance evaluations and comparisons for error-correcting systems employing SBC-BCH codes with the proposed decoding algorithm. The performance evaluations show that the proposed OSA-aided IHDD considerably improves the error-rate performance in both the waterfall and error-floor regions.