Fast forward several years and I'm finally getting around to finishing this quilt. I began by creating blocks of uniform size, either by cutting some panels smaller or adding strips to other panels. I laid out all the panels in chronological order, made sure I had an even number of squares, and stitched them together.
An old sheet sufficed for the backing. While attaching said sheet, I discovered that my top part was not entirely square. Not having any place to trim without damaging the logos, I did what I could and now have a usable display quilt instead of a box full of t-shirts. I don't know that I'll use the quilt or hang it on the wall or do anything other than put it into a memory box, but somehow it seems less wasteful and more nostalgic to have a quilt in storage than to have a bunch of shirts sitting unused.