Not a UPS employee, just a customer who ships a lot and receives a lot.
When your package (tracking) shows it arrived at your local center, make the request for Access point pickup immediately.
Don't wait until your package is sorted and loaded.
Different trucks get different loads (commercial, residential, UPS center transfers, etc).
If you make the request late, your package will go out to the address on the package if not delivered, it will return to the center at the end of the day and then get loaded the next day on the truck that is going to your Access point.
Remember Access point locations will change the truck it gets loaded.
My San Bernardino bound packages I want to pickup in Los Angeles county, will get a center to center transfer to the different county overnight first and then get loaded in the morning - so be aware and plan accordingly....