What could be causing annoying noise that starts as soon as I start the engine

Have you confirmed that the noise is definitely coming from the engine compartment? Ford products have interior electrical components that stay energized after the car is shut off, and are shut down when the door is opened to exit the vehicle. They designed it as a courtesy, the main one being the radio/sound system; you can stop for a moment, shut off the motor, and listen to the radio As long as you don't open the door. Some of those, possibly an under dash one, can sound as if they are coming from the motor.