There are various ways to do it other than cutting a large square and throwing away quite a bit. I use sticky stabilizer most often and what I like to do is, after I (carefully) tear out the first embroidery is just to stick a smaller piece over the hole I just made. This will work several times, usually. It also works to cut a long strip and just work your way down it as you go. And you can make a "frame" to hoop, that is hooped but then almost the whole hoop area inside is a hole for your sticky stabilizer. I've done this, too, with heavy clear plastic. And this would be a place where the 5 inch hoop would be good for 4 inch designs because you'd have that much left inside your clear plastic (or whatever) that you hooped.
Would it be possible to have a permanent piece of stabilizer (or other thin
fabric) you reuse for the hoop with a hole in the middle over which you
baste the stablizier you want to use in your embroidery design - or would
this mean you would not get enough tension in the hoop.