Let G G be a finite group, and α \alpha a nontrivial character of G G . The McKay graph M ( G , α ) \mathcal {M}(G,\alpha ) has the irreducible characters of G G as vertices, with an edge from χ 1 \chi _1 to χ 2 \chi _2 if χ 2 \chi _2 is a constituent of α χ 1 \alpha \chi _1 . We study the diameters of McKay graphs for finite simple groups G G . For alternating groups G = A n G = \mathsf {A}_n , we prove a conjecture made in another work by the authors: there is an absolute constant C C such that d i a m M ( G , α ) ≤ C log ⁡ | G | log ⁡ α ( 1 ) \mathrm {diam}\,{\mathcal M}(G,\alpha ) \le C\frac {\log |G|}{\log \alpha (1)} for all nontrivial irreducible characters α \alpha of G G . Also for classical groups of symplectic or orthogonal type of rank r r , we establish a linear upper bound C r Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a product χ 1 χ 2 ⋯ χ l \chi _1\chi _2\cdots \chi _l of irreducible characters of some finite simple groups G G to contain all irreducible characters of G G as constituents.