This is follow-up to <a href="https://mathoverflow.net/questions/286316/recognition-of-symmetric-groups-in-gap">my earlier question</a>.

Suppose that we have elements $\sigma_1,\ldots,\sigma_k\in S_n$, and that we established that these elements actually generate $S_n$. 
Since that previous question, I learned that the <a href="https://www.sciencedirect.com/science/article/pii/S074771719990295X">Bratus-Pak algorithm</a> for recognising symmetric groups actually also produces a way to express every element of $S_n$ as a word in sigmas. 

However, in some specific applications I have in mind (where $n$ is about 60), the transpositions $(i,j)$ in $S_n$ are thus expressed as words of length about 800 in sigmas, and I have strong reasons to suspect that there are much shorter ways to represent them (shorter than 100). Are there some methods that are known [in practice] to produce reasonably short words in the word group in such situation?  

EDIT: in principle, I can even reduce my problem a little bit, and ask just for a representatives in cosets of transpositions modulo $H$, where $H$ is some explicitly given subgroup of $S_n$. Maybe that could help.