Electrolyte in a rechargeable Li-ion battery plays a critical role in determining its capacity and efficiency. While the typically used electrolytes in Li-ion batteries are liquid, soft solid electrolytes are being increasingly explored as an alternative due to their advantages in terms of increased stability, safety and potential applications in the context of flexible and stretchable electronics. However, ionic conductivity of solid polymer electrolytes is significantly lower compared to liquid electrolytes. In a recent work, we developed a theoretical framework to model the coupled deformation, electrostatics and diffusion in heterogeneous electrolytes and also established a simple homogenization approach for the design of microstructures to enhance ionic conductivity of composite solid electrolytes. Guided by the insights from the theoretical framework, in this paper, we examine specific microstructures that can potentially yield significant improvement in the effective ionic conductivity. We numerically implement our theory in the open source general purpose finite element package FEniCS to solve the governing equations and present numerical solutions and insights on the effect of microstructure on the enhancement of ionic conductivity. Specifically, we investigate the effect of shape by considering ellipsoidal inclusions. We also propose an easily manufacturable microstructure that increases the ionic conductivity of the composite electrolyte by 40 times, simply by the addition of dielectric columns parallel to the solid electrolyte phase.