An important prerequisite for safe control is aligning the policy with the underlying constraints in the environment. In many real-world applications, due to the difficulty of manually specifying these constraints, existing works have proposed recovering constraints from expert demonstrations by solving the Inverse Constraint Learning (ICL) problem. However, ICL is inherently ill-posed, as multiple constraints can equivalently explain the experts' preferences, making the optimal solutions not uniquely identifiable. In this work, instead of focusing solely on a single constraint, we propose the novel approach of Exploratory ICL (ExICL). The goal of ExICL is to recover a diverse set of feasible constraints, thereby providing practitioners the flexibility to select the most appropriate constraint based on the practical needs of deployment. To achieve this goal, we design a generative diffusion verifier that guides the trajectory generation process using the probabilistic representation of an optimal constrained policy. By comparing these decisions with those made by expert agents, we can efficiently verify a candidate constraint. Driven by the verification feedback, ExICL implements an exploratory constraint update mechanism that strategically facilitates diversity within the collection of feasible constraints. Our empirical results demonstrate that ExICL can seamlessly and reliably generalize across different tasks and environments. The code is available at https://github.com/ZhaoRunyi/ExICL.