In 1694, Gregory and Newton discussed the problem to determine the kissing number of a rigid material ball. This problem and its higher dimensional generalization have been studied by many mathematicians, including Minkowski, van der Waerden, Hadwiger, Swinnerton-Dyer, Watson, Levenshtein, Odlyzko, Sloane and Musin. Recently, Li and Zong introduced and studied the generalized kissing numbers of convex bodies. As a continuation of this project, in this paper we obtain the exact generalized kissing numbers $κ_α^*(B^n)$ of the $n$-dimensional balls for $3\le n\le 8$ and $α=2\sqrt{3}-2$. Furthermore, the lattice kissing number of a four-dimensional cross-polytope is determined.