In earlier papers, we showed a decomposition of 2-diregular digraphs (2-dds) and used it to provide some sufficient conditions for these graphs to be non-Hamiltonian; we also showed a close connection between the permanent and determinant of the adjacency matrices of these digraphs and gave some enumeration and generation results. In the present paper we extend the discussion to a larger class of digraphs, introduce the notions of routes and quotients and use them to provide additional criteria for 2-dds to be non-Hamiltonian. Though individual non-Hamiltonian regular connected graphs of low degree are known (e.g. Tutte and Meredith graphs), families of such graphs are not common in the literature; even scarcer are families of such digraphs. Our results identify a few such families.