In Part I, a generalized finite-volume direct averaging micromechanics (FVDAM) theory was constructed for periodic materials with complex microstructures undergoing finite deformations. The generalization involves the use of a higher-order displacement field representation within individual subvolumes of a discretized analysis domain whose coefficients were expressed in terms of surface-averaged kinematic variables required to be continuous across adjacent subvolume faces. In Part II of this contribution we demonstrate that the higher-order displacement representation leads to a substantial improvement in subvolume interfacial conformability and smoother stress distributions relative to the original theory based on a quadratic displacement field representation, herein called the order theory. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The advantages of the generalized theory are illustrated through examples based on a known analytical solution and finite-element results generated with a computer code that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the order theory that revealed microstructural effects in this class of materials which are important in bio-inspired material architectures that mimic certain biological tissues.