In a previous publication [Betzinger, Friedrich, Görling, and Blügel, Phys. Rev. B 85, 245124 (2012)] we presented a technique to compute accurate all-electron response functions, e.g., the density response function, within the full-potential linearized augmented-plane-wave (FLAPW) method. Response contributions that are not captured (completely) within the finite Hilbert space spanned by the LAPW basis are taken into account by an incomplete-basis-set correction (IBC). The latter is based on a

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... formal response of the basis functions themselves, which is derived by exploiting their dependence on the effective potential. Its construction requires the solution of radial differential equations, having the form of Sternheimer equations, by numerical integration. The approach includes a formally exact treatment of the response contribution from the core states. While we restricted the formalism to spherical perturbations in the previous work, we here generalize the formalism to nonspherical perturbations. The improvements are demonstrated with exact-exchange optimized-effective-potential (EXX-OEP) calculations of antiferromagnetic NiO. It is shown that with the generalized IBC a basis-set convergence is realized that is as fast as in density-functional theory calculations using standard local or semilocal functionals. The EXX-OEP band gap, magnetic moment, and spectral function of NiO are in substantially better agreement with experiment than results obtained from calculations with local and semilocal functionals.