Staircase codes (SCCs) are typically decoded using iterative bounded-distance decoding (BDD) and hard decisions. In this paper, a novel decoding algorithm is proposed, which partially uses soft information from the channel. The proposed algorithm is based on marking certain number of highly reliable and highly unreliable bits. These marked bits are used to improve the miscorrection-detection capability of the SCC decoder and the error-correcting capability of BDD. For SCCs with 2-error-correcting Bose-Chaudhuri-Hocquenghem component codes, our algorithm improves upon standard SCC decoding by up to 0.30 dB at a bit-error rate (BER) of 10-7. The proposed algorithm is shown to achieve almost half of the gain achievable by a genie decoder with this structure. The increased complexity caused by bit marking and additional calls to the component BDD decoder is discussed as well. Our algorithm is also extended (with minor modifications) to product codes. The simulation results show that in this case, the algorithm offers gains of up to 0.5 dB at a BER of 10-7.