Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://anonymous.4open.science/r/SOC-ICNN-4B18/.