Airtag and iPhone: last known location

I often forget where I've parked, so I'd like to try AirTag/Find My, with the AirTag in the glove compartment.

I understand last known location, but I want to know whether that location is "set" if the phone is On but not awake. For instance, the phone hasn't been used all day, I get out of my car at a store parking lot and then walk out of the AirTag's range. Has Find My recorded that the car/AirTag is in the parking lot? Is that the last known location?

IOW, does the phone need to be awake to record where the AirTag was last located?

Thanks.