As learning-based methods for legged robots rapidly grow in popularity, it is important that we can provide safety assurances efficiently across different controllers and environments. Existing works either rely on a priori knowledge of the environment and safety constraints to ensure system safety or provide assurances for a specific locomotion policy. To address these limitations, we propose an observation-conditioned reachability-based (OCR) safety-filter framework. Our key idea is to use an OCR value network (OCR-VN) that predicts the optimal control-theoretic safety value function for new failure regions and dynamic uncertainty during deployment time. Specifically, the OCR-VN facilitates rapid safety adaptation through two key components: a LiDAR-based input that allows the dynamic construction of safe regions in light of new obstacles and a disturbance estimation module that accounts for dynamics uncertainty in the wild. The predicted safety value function is used to construct an adaptive safety filter that overrides the nominal quadruped controller when necessary to maintain safety. Through simulation studies and hardware experiments on a Unitree Go1 quadruped, we demonstrate that the proposed framework can automatically safeguard a wide range of hierarchical quadruped controllers, adapts to novel environments, and is robust to unmodeled dynamics without a priori access to the controllers or environments - hence, ``One Filter to Deploy Them All''.