The back came from the same piece of maple as the neck and sides. I cut the stock into 5 inch wide boards, each about 5/16ths of an inch thick. This was a bit thicker than needed, but I didn't know how much planing would remove (and too thick is always better than too thin.) After planing to remove the rough bandsaw cut edge, the thickness was about 3/16ths thick.
Most important while jointing the back by hand (especially with a very short plane, a longer plane would make this much easier) was to not over do it. I started by taking the two ends down since they are waste and so they wouldn't interfere with the center section which needs to fit precisely. For the actual jointing, I started with the edges matched together as closely as possible. Then I ran the plane over the edges in a few smooth strokes, just to bring them flush and parallel to one another. This all that was needed to get a near invisible joint.
Glueing the Center Joint
To put even tension on the joint, I secured long wooden braces on either side of the back a little closer together than necessary for the back to fit so that the back would tee-pee. With the clamps secured and glue added to the joint, I collapsed the tee-pee and added a clamp down the center to hold everything flat.
Glueing on the Bracing
I didn't join the lateral braces to the center support. Instead, the center support is made from three different pieces. Very soon after I attached the bracing, the back quickly bowed. I'm still not sure why this happened. It wasn't so much of a problem that I couldn't fit the back to the rim, but it was annoying. It happened again on my second mandolin (a similar design), so I removed the cross braces leaving only the vertical center brace (which is one piece on that mandolin.) I doubt the lack of lateral bracing will hurt the sound, and I know it is still plenty strong. I am definetely just going leave the cross braces off my next flat top entirely.