The Simplex Architecture ensures the safe use of an unverifiable complex controller by using a verified safety controller and verified switching logic. This architecture enables the safe use of high-performance, untrusted, and complex control algorithms without requiring them to be formally verified. Simplex incorporates a supervisory controller and safety controller that will take over control if the unverified logic misbehaves. The supervisory controller should (1) guarantee the system never enters and unsafe state (safety), but (2) use the complex controller as much as possible (minimize conservatism). The problem of precisely and correctly defining this switching logic has previously been considered either using a control-theoretic optimization approach, or through an offline hybrid systems reach ability computation. In this work, we prove that a combined online/offline approach, which uses aspects of the two earlier methods along with a real-time reach ability computation, also maintains safety, but with significantly less conservatism. We demonstrate the advantages of this unified approach on a saturated inverted pendulum system, where the usable region of attraction is 227% larger than the earlier approach.