3v3 Microcontroller controlling a MOSFET that needs 5V
What you probably want to do is add another small transistor (NPN or NMOS) in an open-drain configuration as a buffer. So what you would do is add a pull up resistor to the gate on your current transistor and then use the new transistor to pull the gate to ground to turn it off. You will also want to add a pull up to the new transistor as well so that it turns off the power transistor by default. You can add a Zener diode to limit the gate voltage. The control voltage will end up being inverted, but you should be able to turn the transistor all the way on.