Model reduction based on matrix interpolation provides an efficient way to compute the dynamic response of nearly periodic structures composed of substructures (cells) with varying properties. This may concern 2D or 3D substructures subjected to geometric modifications (mesh variations) or more classic dimension changes. An efficient interpolation strategy for a nearly periodic structure can be obtained by (i) interpolating reduced substructure matrices over a multi-dimensional parametric space and (ii) reducing the number of degrees of freedom at substructure boundaries by considering the interface modes of an equivalent purely periodic structure. In this paper, two basis enrichment techniques are proposed to improve the accuracy of the interpolation strategy. This consists in (i) considering high-order static modes, in addition to component modes, to express the reduced substructure matrices and (ii) adding static correction vectors into the basis of interface modes to account for the varying properties of the substructures. Numerical experiments are carried out which clearly highlight the relevance of the basis enrichment techniques for predicting the harmonic behavior of nearly periodic structures with 2D or 3D substructures.