The generating graph of some monolithic groups

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We look for conditions on the positive integer m that ensure that Γ(G) contains a Hamiltonian cycle when G=S≀Cm is the wreath product of a finite simple group S and a cyclic group of order m.