In the literature, a class of iterative decoding algorithms which combine the traditional reliability-based decoding (RBD) with the adaptive belief propagation (ABP) have been validated to be applicable for Reed-Solomon codes. However, in the original design of the iterative decoding, the soft-information is passed only from the ABP-part to the RBD-part such that the decoding performance is somewhat limited. In this study, we first present a new reliability updating scheme for the bidirectional exchange of soft-information in the iterative decoding, which can guarantee the correction of the most errors in both of the reliable and unreliable bits. A simple bit-flipping mechanism is also proposed to refine the initialization of the ABP-part for further performance improvement. Revealed by the simulation results, our proposed scheme can outperform the conventional design in terms of the bit-error-rate performance.