This paper proposes methods for set-valued state estimation of nonlinear, discrete-time systems with nonlinear measurements. This is achieved by combining a novel construct called the input-output set, which approximates system dynamics and measurements, with the hybrid zonotope set representation that can efficiently represent nonconvex and disjoint sets. The methods leverage existing approaches for approximation of nonlinear functions including special ordered set approximations and neural networks. A numerical example demonstrates the proposed method with comparison to a convex approach.