The coherent potential approximation has historically allowed the efficient study of disorder effects over a variety of solid state systems. Its original formulation is, however, limited to a single-site or uncorrelated model of local substitutions. This neglects the effects of correlation and short-range ordering, often found in real materials. Recent theoretical work has shown one possible way to systematically address such shortcomings for simple materials with only one element per unit cell. We briefly review the basic ideas of such development within the framework of multiple scattering theory and suggest its generalization to materials with complex lattices. We validate our extension through a systematic comparison with a classic Cu_1−cZn−c reference test case and propose, for further illustration of local environment effects, the example of the yttria-stabilized cubic phase of zirconia, re-examined through various techniques for the first-principles treatment of disorder.