  Product codes (PCs) and staircase codes (SCCs) are conventionally decoded
based on bounded distance decoding (BDD) of the component codes and iterating
between row and column decoders. The performance of iterative BDD (iBDD) can be
improved using soft-aided (hybrid) algorithms. Among these, iBDD with combined
reliability (iBDD-CR) has been recently proposed for PCs, yielding sizeable
performance gains at the expense of a minor increase in complexity compared to
iBDD. In this paper, we first extend iBDD-CR to SCCs. We then propose two novel
decoding algorithms for PCs and SCCs which improve upon iBDD-CR. The new
algorithms use an extra decoding attempt based on error and erasure decoding of
the component codes. The proposed algorithms require only the exchange of hard
messages between component decoders, making them an attractive solution for
ultra high-throughput fiber-optic systems. Simulation results show that our
algorithms based on two decoding attempts achieve gains of up to $0.88$ dB for
both PCs and SCCs. This corresponds to a $33\%$ optical reach enhancement over
iBDD with bit-interleaved coded modulation using $256$ quadrature amplitude
modulation.
