How to override the automatic heater?

Yep, to override the automatic heater in a 2003 Mercury Grand Marquis, you can do the following:

Turn on the ignition without starting the engine. Press and hold the "off" and "floor" buttons on the climate control panel at the same time. After about 2 seconds, the climate control panel will display "00". Continue holding the buttons for another 2 seconds until the display changes to "01". Release the "off" and "floor" buttons, and then press the "auto" button. The climate control system will now operate in manual mode. In manual mode, you can adjust the temperature, fan speed, and air flow direction using the corresponding buttons on the climate control panel. To return to automatic mode, simply press the "auto" button again.

I would wait until the engine is cold and then check the coolant level which is the #1 reason a heater will stop working. please check out this guide below to help you fix the problem as well:

https://www.2carpros.com/articles/car-heater-not-working

Please go over this guide and report back.

Okay, I tried it a bunch of times with no results. The heat works when it kicks on. When it kicks on, it works fantastically. The car has a lot of miles and years but was very well cared for and there was only one owner for all the miles. I was wondering if what reads the outside temperature is the problem. The display reads warmer than it is. I'd say 10 to 20 degrees warmer.

Yep, that can do it, the misinformation can fool the HVAC controller, here is the location of the ambient temperature sensor in the front of the car and the instructions on how to change it out.