We considered a self-diffeomorphism of the plane with a fixed hyperbolic point at the origin and a non-transverse point homoclinic to it. Periodic points located in a sufficiently small neighborhood of the homoclinic point are divided into single-pass and multi-pass points depending on the location of the orbit of the periodic point with respect to the orbit of the homoclinic point. It follows from the works of W. Newhouse, L.P. Shil’nikov, B.F. Ivanov and other authors that for a certain method of tangency of the stable and unstable manifolds there can be an infinite set of stable periodic points in a neighborhood of a non-transverse homoclinic point, but at least one of the characteristic exponents of these points tends to zero with increasing period. Previous works of the author imply that for a different method of tangency of the stable and unstable manifolds there can be an infinite set of stable single-pass periodic points, the characteristic exponents of which are bounded away from zero in the neighborhood of a non-transverse homoclinic point. It is shown in this paper that under certain conditions imposed primarily on the method of tangency of the stable and unstable manifolds there can be a countable set of two-pass stable periodic points, the characteristic exponents of which are bounded away from zero in any neighborhood of a non-transverse homoclinic point.