Question About Redoing AF Quests

How does this work if you upgrade a piece of AF to AF +1. Can you redo just the quest for that piece and get it so you can store the set? Or do you have to throw all of your AF away and start from the first piece?

You don't have to throw anything away to start re-questing it; it's just at the end that still holding the original piece poses problems, as you can't re-obtain it while holding it, and hence, can't flag the quest as finished. Remember, you need to re-quest the weapon first.

Note that if your AF is in storage and you erase your memory and re-quest it, then you can still re-obtain the AF; you just won't be able to take your AF out of storage while holding onto said piece.