A cork board is the first thing I think of for easiest accessibility (and you could have a little bowl of backs next to it or even tacked onto the board) but personally, (even if a smidge less accessible) I have a few hanger jewelry organizers that I LOVE and have worked great with my huge jewelry collection. Like this. There are a lot of versions, including ones with stud earring organizers, and I have a few that zip up. I like the transparency so it’s easy for me to see everything (not just earrings) as I flip through my closet.

Edit: I also think you could easily, like others have suggested, slide little cardboard/thick paper inserts in the clear pockets for the earring studs. Now that I’ve seen that here I may just do it myself.