Hamilton-Jacobi (HJ) reachability provides formal safety guarantees for dynamical systems, but solving high-dimensional HJ partial differential equations limits its use in real-time planning. This paper presents a contingency-aware multi-goal navigation framework that integrates learning-based reachability with sampling-based planning in unknown environments. We use Fourier Neural Operator (FNO) to approximate the solution operator of the Hamilton-Jacobi-Isaacs variational inequality under varying obstacle configurations. We first provide a theoretical under-approximation guarantee on the safe backward reach-avoid set, which enables formal safety certification of the learned reachable sets. Then, we integrate the certified reachable sets with an incremental multi-goal planner, which enforces reachable-set constraints and a recovery policy that guarantees finite-time return to a safe region. Overall, we demonstrate that the proposed framework achieves asymptotically optimal navigation with provable contingency behavior, and validate its performance through real-time deployment on KUKA's youBot in Webots simulation.