We investigate the asymptotic properties of higher-order binding corrections to the one-loop self-energy of excited states in atomic hydrogen. We evaluate the historically problematic A60 coefficient for all P states with principal quantum numbers n</=7 and D states with n</=8 and find that a satisfactory representation of the n dependence of the coefficients requires a three-parameter fit. For the high-energy contribution to A60, we find exact formulas. The results obtained are relevant for the interpretation of high-precision laser spectroscopic measurements.