Abstract: Let $\mbox{Sym}(n)$ be the symmetric group on $\{1,\dots,n\}$ and $\mathcal{T}$ be a set of some transpositions of $\mbox{Sym}(n)$. Let $G(\mathcal{T})$ be the graph with vertex set $\{1,\dots,n\}$ such that there is an edge $ij$ in $G(\mathcal{T})$ if and only if the transposition $[i,j]\in\mathcal{T}$. In this paper we show that, for $n\geq 4$, the generalized $3$-connectivity of the Cayley graph on $\mbox{Sym}(n)$ generated by $\mathcal{T}$ is $n-1$ if $G(\mathcal{T})$ is any unicyclic graph.