  We revisit and sharpen the results from our previous work, where we
investigated the regularity of the singular set of the free boundary in the
nonlinear obstacle problem. As in the work of Figalli-Serra on the classical
obstacle problem, we show that each stratum can be further decomposed into a
`good' part and an `anomalous' part, where the former is covered by $C^{1,1-}$
manifolds, and the latter is of lower dimension.
