My "butterfly chair" had gotten kind of ragged. I got it at Target when I was in high school, and after 3 or for 4 years of using it at home and then 4 more years of using it at college, it needed an upgrade. The frame was still good, but the fabric part was shot. How hard could it be to just sew a new one? I thought. This whole project was an improvisation, but it turned out pretty well.
The main issue with the old cover was that, where the cover fit over the frame, the seams had ripped over time and the frame was poking through. So, in my version, I tried not to have seams at those high impact areas of the cover - folding it over instead, so the fabric would receive most of the force (rather than a seam of thread), and it would be distributed over a wider area. I also build into two "pockets" that fit over the frame - the main one, made of the bottom fabric (duck cloth), and also a backup, made of the top fabric (upholstery fabric).
I couldn't resist personalizing it by sewing the outline of a seriffed "E" into the back. Then, I used bias tape to finish the edges. The two main cats in my life each gave it their seal of approval!