The Korringa-Kohn-Rostoker (KKR) Green's function, multiple-scattering theory is an efficient site-centered, electronic-structure technique for addressing an assembly of N scatterers. Wave functions are expanded in a spherical-wave basis on each scattering center and indexed up to a maximum orbital and azimuthal number Lmax=(l,m)max, while scattering matrices, which determine spectral properties, are truncated at Ltr=(l,m)tr where phase shifts δl>ltr are negligible. Historically, Lmax is set equal to Ltr, which is correct for large enough Lmax but not computationally expedient; a better procedure retains higher-order (free-electron and single-site) contributions for Lmax>Ltr with δl>ltrset to zero [X.-G. Zhang and W. H. Butler, Phys. Rev. B 46, 7433 (1992)]. We present a numerically efficient and accurate augmented-KKR Green's function formalism that solves the KKR equations by exact matrix inversion [R3 process with rank N(ltr+1)2] and includes higher-L contributions via linear algebra [R2 process with rank N(lmax+1)2]. The augmented-KKR approach yields properly normalized wave functions, numerically cheaper basis-set convergence, and a total charge density and electron count that agrees with Lloyd's formula. We apply our formalism to fcc Cu, bcc Fe, and L10CoPt and present the numerical results for accuracy and for the convergence of the total energies, Fermi energies, and magnetic moments versus Lmax for a given Ltr.