Others have answered in equations; let me see if I can put the answer in words, for possibly easier understanding. When ball hits bat, its kinetic energy is (with some reasonably high efficiency) converted into potential energy by the ball deforming from its equilibrium shape; then that gets reconverted to kinetic energy (now going in the opposite direction) as the ball springs back to roundness. In other words, the ball bounces. (If you assume a rigid ball the problem changes a lot.) So we can rephrase the question: Which will bounce higher, a ball that’s just dropped from some given height, or a ball that is thrown downward so it hits the ground at a higher speed? Now we intuitively see that it is the latter. Bouncing from a moving bat is no different from bouncing off the ground.
Now there are some caveats to that; the efficiency of the energy conversion is not constant, it’s some function of the ball’s speed—at some point the ball doesn’t bounce at all, it just disintegrates (or bursts into flames, if the loss to heat is sufficient). But for speeds achievable by human throwing arms, this effect can probably be ignored.
Others have answered in equations; let me see if I can put the answer in words, for possibly easier understanding. When ball hits bat, its kinetic energy is (with some reasonably high efficiency) converted into potential energy by the ball deforming from its equilibrium shape; then that gets reconverted to kinetic energy (now going in the opposite direction) as the ball springs back to roundness. In other words, the ball bounces. (If you assume a rigid ball the problem changes a lot.) So we can rephrase the question: Which will bounce higher, a ball that’s just dropped from some given height, or a ball that is thrown downward so it hits the ground at a higher speed? Now we intuitively see that it is the latter. Bouncing from a moving bat is no different from bouncing off the ground.
Now there are some caveats to that; the efficiency of the energy conversion is not constant, it’s some function of the ball’s speed—at some point the ball doesn’t bounce at all, it just disintegrates (or bursts into flames, if the loss to heat is sufficient). But for speeds achievable by human throwing arms, this effect can probably be ignored.