We compute the third-order correction to the S-wave quarkonium wave functions
|\psi_n(0)|^2 at the origin from non-Coulomb potentials in the effective
non-relativistic Lagrangian. Together with previous results on the Coulomb
correction and the ultrasoft correction computed in a companion paper, this
completes the third-order calculation up to a few unknown matching
coefficients. Numerical estimates of the new correction for bottomonium and
toponium are given.