  In 2020, Siemons and Zalesski [On the second eigenvalue of some Cayley graphs
of the symmetric group. {\it arXiv preprint arXiv:2012.12460}, 2020] determined
the second eigenvalue of the Cayley graph $\Gamma_{n,k} =
\operatorname{Cay}(\operatorname{Sym}(n), C(n,k))$ for $k = 0$ and $k=1$, where
$C(n,k)$ is the conjugacy class of $(n-k)$-cycles. In this paper, it is proved
that for any $n\geq 3$ and $k\in \mathbb{N}$ relatively small compared to $n$,
the second eigenvalue of $\Gamma_{n,k}$ is the eigenvalue afforded by the
irreducible character of $\operatorname{Sym}(n)$ that corresponds to the
partition $[n-1,1]$. As a byproduct of our method, the result of Siemons and
Zalesski when $k \in \{0,1\}$ is retrieved. Moreover, we prove that the second
eigenvalue of $\Gamma_{n,n-5}$ is also equal to the eigenvalue afforded by the
irreducible character of the partition $[n-1,1]$.
