Author’s summary: A corollary to the known classification of finite simple groups having chain difference one [B. Brewster, M. B. Ward and I. Zimmermann, J. Algebra 160, No. 1, 179-191 (1993; Zbl 0791.20016)] is that the length of such groups is either four or five. We prove a partial converse of the corollary to obtain a more precise description of these groups, the number of which depends on the unsolved prime \(k\)-tuples conjecture.