An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection of n spanning subgraphs of K_n, all isomorphic to G, such that any two members of share exactly one edge and every edge of K_n is contained in exactly two members of . In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of length n-1. It is known that the existence of an ODC of K_n by a Hamiltonian path implies the existence of ODCs of K_4n and of K_16n, respectively, by almost-Hamiltonian cycles. Horton and Nonay introduced two-colorable ODCs and showed: If there are an ODC of K_n by a Hamiltonian path for some n3 and a two-colorable ODC of K_q by a Hamiltonian path for some prime power q5, then there is an ODC of K_qn by a Hamiltonian path. In [U. Leck, A class of 2-colorable orthogonal double covers of complete graphs by hamiltonian paths, Graphs Combin. 18 (2002) 155–167], two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths were constructed for all odd square numbers n9. Here we continue this work and construct cyclic two-colorable ODCs of K_n and K_2n, respectively, by Hamiltonian paths for all n of the form n=4k²+1 or n=(k²+1)/2 for some integer k.