How to completely unlock a hat objective?

Usually getting the "unlock a hat" means that you need to collect all the types of artifacts of that type and then you get the "unlock a hat" message and the achievement, if needed, as marked as complete. I don't think you need to wear the hat. If you got all of one type (rings, masks, critters, floral or relics, or one of the holiday ones) then you should have completed the achievement. It used to be that getting the last one of a type (the tenth ring, for example) would take quite a long time, but one of the updates in the last 6 months apparently adjusted the probabilities to make it more likely to get things you don't already have.