Let \(A\) be a finite abelian group, \(\Delta\subseteq A\), and \(\varphi\) an endomorphism of \(A\). A digraph whose vertices are the elements of \(A\) and whose arcs are the pairs \((x,\varphi(x)+a)\) with \(x\in A\) and \(a\in A\) is called an endo-Cayley digraph. In this paper Hamiltonicity properties of endo-Cayley digraphs are investigated. The main results concern consecutive digraphs, i.e., endo-Cayley digraphs with a finite cyclic group \(A\) as set of vertices and \(\Delta=\{e,e+h,e +2h,\dots\}\). The authors develop a line graph technique which can be applied to find sufficient conditions for Hamiltonicity. Moreover, it is shown that results as the factor group lemma (known for Cayley digraphs on abelian groups) can also be proved for endo-Cayley graphs.