In this paper, we improve the forward-backward algorithm for quasi-linear PDEs introduced in a previous work. The new discretization scheme takes advantage of the standing regularity properties of the true solution through an interpolation procedure. For the convergence analysis, we also exploit the optimality of the square Gaussian quantization used to approximate the conditional expectations involved. The resulting bound for the error is closely related to the Holder exponent of the second order spatial derivatives of the true solution and turns out to be more satisfactory than the one previously established.