I was reading about the propulsive efficiency of jet engines earlier, when I thought about the Oberth Effect. A jet (or any type of reaction engine) is most efficient when operating at the velocity equal to that of its exhaust; this is because if the exhaust is traveling at zero velocity in the vehicle's frame of reference, then the vehicle gains all the kinetic energy from the propellant and the exhaust is left with no kinetic energy. If a rocket were to be traveling faster than its exhaust velocity, however, then the exhaust would still have some of the kinetic energy that the rocket's propellant originally had, which would result in lower propulsive efficiency.
Why then does the Oberth Effect still increase the available kinetic energy when rocket velocity exceeds exhaust velocity?