Let $X$ be a subset of $S_n$. I suppose that the size of $X$ is small compared to the order of $S_n$. In order to quickly check whether $X$ generates $S_n$ I would do the following: For $k=1,2,3\dots$ check whether the group generated by $X$ is $k$-fold transitive. That is a cheap test: Let $\Gamma$ be the graph whose vertices are the $k$-tuples of distinct elements from $\{1,2,\dots,n\}$. Connect two vertices by an edge if an element from $X$ moves one vertex to the other one. Then the group generated by $X$ is $k$-transitive if and only if $\Gamma$ is connected, which is algorithmically easy and cheap to test.

If $X$ passes the test up to $k=6$, then you know that the generated group is $A_n$ or $S_n$, because there are no other $6$-transitive groups. Deciding between these two cases is a matter of checking the signum of the elements from $X$.

In most degrees $n$, there are no $2$-transitive groups besides $A_n$ or $S_n$, so of course you can stop with $k=2$.