Yes, $(S_n, E_n)$ contains a Hamiltonian cycle for every $n \in \mathbb{N}$.  This follows by the [Steinhaus–Johnson–Trotter algorithm
][1].  The transpositions can even be chosen to be consecutive elements in the previous permutation.  


  [1]: https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm