A directed Toeplitz graph is a digraph with a Toeplitz adjacency matrix. In this paper we contribute to [6]. The paper [6] investigates the hamiltonicity of the directed Toeplitz graphs \(T_n\langle s_1,s_2,…, s_k;t_1, t_2,…,t_l\rangle\) with \(s_2 = 2\) and in particular those with \(s_3 = 3\). In this paper we extend this investigation to \(s_2 = 3\) with \(s_1 =t_1 =1\).