The author improves a marching method for the numerical solution of a generalized (non-convex) eikonal equation in \(\mathbb R^N\) considered earlier by the author and co-workers [see E. Carlini, M. Falcone, N. Forcadel, and R. Monneau, ibid. 46, No. 6, 2920–2952 (2008; Zbl 1180.65112)]. He shows by a counter-example that the earlier method is not monotone in case the normal velocity changes sign and proposes a modified version (on a square grid, using a time step not restricted by a Courant-Friedrichs-Lewy (CFL) condition) which is shown to possess a comparison principle from which monotonicity follows. He also proves the convergence of the modified version and provides a heuristic error estimate.