The results of a numerical implementation of the recent average atom model including ion-ion correlations of Starrett and Saumon [Phys. Rev. E 85, 026403 (2012)] are presented. The solution is obtained by coupling an average atom model to a two-component plasma model of electrons and ions. The two models are solved self-consistently and results are given in the form of pair distribution functions. Ion-ion pair distribution functions for hydrogen, carbon, aluminum, and iron are compared to quantum and Thomas-Fermi molecular dynamics simulations as well as path-integral Monte Carlo calculations and good agreement is found for a wide variety of plasma conditions in the warm and hot dense matter regime.