This paper delivers an overview of finite element analysis (FEA) of contact problems involving non-conforming meshes (NCM) in the perspective of small-deformation inelasticity. FEA of solid mechanics commonly utilize NCM to enhance the exactness in simulating the actual nonlinear behaviors without the need for added computational time and cost. Materials nonlinearity, plasticity, higher-order variables are important and sensitive issues that require careful considerations to effectively characterize the nonlinear behavior and deformation history across two non-matching boundaries. The Enriched Discontinuous Galerkin Approach (EDGA) is capable of delivering interfacial higher-order kinematic fields, but the matter of preserving material history remains problematic at the integration points for a needed higher-order integration. Accordingly, the EDGA is expanded to entail small deformation plasticity cases by employing a contact-driven Gauss-Kronrod (GK) integration regulation to permit dynamic interfacial enrichment with preserved the history-dependent data at prevailing integration points. This paper presents unique case studies and applications utilizing the traditional J2 plasticity theorem in order to validate the viability and efficiency in refining algorithmic performance and to provide a reliable methodology for the coupled NCM in plasticity.