This work focuses on the LDPC codes for the packet erasure channel, also called AL-FEC (Application-Level Forward Error Correction codes). Previous work has shown that the erasure recovery capabilities of LDPC-triangle and LDPC-staircase AL-FEC codes can be greatly improved by means of a Gaussian Elimination (GE) decoding scheme, possibly coupled to a preliminary Zyablov Iterative Decoding (ID) scheme. Thanks to the GE decoding, the LDPC-triangle codes were very close to an ideal code. If the LDPC-staircase performances were also improved, they were not as close to an ideal code as the LDPC-triangle codes were. The first goal of this work is to reduce the gap between the LDPC-staircase codes and the theoretical limit. We show that a simple modification of the parity check matrix can significantly improve their recovery capabilities when using a GE decoding. Unfortunately the performances of the same codes featuring an ID are negatively impacted, as well as the decoding complexity. The second goal of this work is therefore to find an appropriate balance between all these aspects.