The diffusion of tracer particles within a polymer environment is a promising topic connected with numerous biological and industrial applications, including intracellular macromolecule transport and nanoparticle diffusion. Despite extensive studies, the study of self-propelled particles within a polymer network has received attention recently and poorly understood. Here we study the nonequilibrium diffusion of active tracers navigating through a polymer network. It is shown that active tracers escape the confined geometry with their self-propulsion activity, performing activity-induced hopping diffusion distinguished from that from thermal energy. We investigate how the active hopping diffusion is characterized by physical conditions such as the mesh-to-particle size ratio, bending stiffness of a meshwork, and the Peclet number. We provide a first-passage time theory of active escaping phenomena to explain the observed trapped times of active tracers within a meshwork. Finally, we extend our analysis to randomly cross-linked polymer networks. Through quantitative investigation, we elucidate how heterogeneous non-Gaussian active diffusion emerges in such random polymer networks, shedding light on the complex interplay between polymer structure and active particle dynamics.