This paper studies orthogonal double covers of the complete graph by almost Hamiltonian cycles, i.e. decompositions of \(2K_n\) by cycles of length \(n-1\) which pairwise share exactly one edge. Such covers were known to exist for \(n\) and odd prime power [B. Alspach, K. Heinrich and M. Rosenfeld, Isr. J. Math. 40, 118-128 (1981; Zbl 0472.05027)]. Here the authors introduce a new class of such decompositions which shows that they exist for \(n=q+1\) when \(q=2^et+1\) is a prime power and \(t\) is large enough. By specializing with \(e=1\), they show that the decompositions exist for \(n=q+1\) and \(q \equiv 3 \pmod 4\) a prime.