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Let $$\Gamma$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$G=\mathrm{Cay}(\Gamma,T)$$ be a Cayley graph of $$\Gamma$$. The graph $$G$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for \textcolor[rgb]{0,0,1}{the second (largest) eigenvalue} of the adjacency matrix of the graph $$G$$ in terms of the second eigenvalues of certain subgraphs of $$G$$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$.
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