My Stick Only Drifts After I Have Been Playing For An Hour Or So

This is a real phenomenon, you have diagnosed it correctly as temperature related, and it tells you something useful about where the controller is in its life.

Two things change as the controller warms. The potentiometer's resistance has a temperature coefficient, so the voltage it returns at a fixed physical position shifts slightly with heat. On a new stick that shift is tiny and hides inside the normal noise band. The second change is mechanical: the plastic housing and the stick gimbal expand, and a gimbal that has worn even slightly now has a little more play to move into. Together they move the reported centre, and the shift grows over the first hour because that is how long the controller takes to reach a stable temperature with your hands on it.

The important part is why a new controller does not do this. On a healthy stick, the thermal shift exists but stays under about 0.03, which is inside the band everything ignores. Your 0.09 means the mechanical slack is already there, and heat is simply revealing it earlier than resting measurements would. So this is not a different fault from ordinary drift. It is ordinary drift, caught early, which is genuinely useful to know.

What that means practically: you have time. Early thermal drift typically progresses to permanent cold drift over several months rather than weeks, and you will see your cold resting figure start to climb above 0.01 as that happens. Keep testing cold once a month and write the number down. When the cold figure passes about 0.05, order the stick module, because that is when it will start affecting play regardless of temperature. Until then there is nothing to fix and nothing to worry about, and raising your in-game deadzone now would cost you precision for a problem you only have after an hour.