This paper presents new achievability bounds on the maximal achievable rate of variable-length stop-feedback (VLSF) codes operating over a binary erasure channel (BEC) at a fixed message size M = 2^k. We provide bounds for two cases: The first case considers VLSF codes with possibly infinite decoding times and zero error probability. The second case limits the maximum (finite) number of decoding times and specifies a maximum tolerable probability of error. Both new achievability bounds are proved by constructing a new VLSF code that employs systematic transmission of the first k message bits followed by random linear fountain parity bits decoded with a rank decoder. For VLSF codes with infinite decoding times, our new bound outperforms the state-of-the-art result for BEC by Devassy et al. in 2016. We show that the backoff from capacity reduces to zero as the erasure probability decreases, thus giving a negative answer to the open question Devassy et al. posed on whether the 23.4% backoff to capacity at k = 3 is fundamental to all BECs. For VLSF codes with finite decoding times, numerical evaluations show that the systematic transmission followed by random linear fountain coding performs better than random linear coding in terms of achievable rates.