The existence of a seam of conical intersection, the reference seam, does not rule out the existence of additional disjoint seams of conical intersection. These disjoint seams intersect the g-h planes of the reference seam, a region usually assumed to be devoid of intersections, potentially leading to unexpected points of degeneracy in close proximity to the original conical intersection. Here the authors show how the locus of these disjoint seams can be predicted employing a Hamiltonian derived from second-order perturbation theory. Dramatic differences between the g-h planes of the reference and disjoint seams are found and are expected to have a profound impact on nuclear dynamics. Numerical studies of both high symmetry (D3h, C3H3) and low symmetry (C2v, C2H2N) species are presented.