Recent results [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are put into the context of earlier work [A. V. Gorbach and M. Johansson, Eur. Phys. J. D 29, 77-93 (2004); M. Johansson and A. V. Gorbach, Phys. Rev. E 70, 057604 (2004)]. The two newly found families of "beyond-band discrete solitons " [M. C. Tran and T. X. Tran, Chaos 32, 073113 (2022)] are found to be smooth continuations of "on-top breathers " briefly mentioned by Gorbach and Johansson [Eur. Phys. J. D 29, 77-93 (2004)] and Johansson and Gorbach [Phys. Rev. E 70, 057604 (2004)] and the relevant bifurcation scenarios are described. One family is shown to be linearly unstable with respect to symmetry-breaking oscillations.