You can make your own voltage regulator with an op-amp driving a MOSFET. Either N-channel or P-channel will work, you would just change the output direction. This solution has no minimum regulator voltage dropout.
If the input is known to be a regulated 12V, you don't need a zener diode to create a known voltage, a resistive divider will do. Compare against the output voltage, and include a cap to slow down the feedback response a lot to prevent oscillation.