I use the leftover bits for making braided rugs (in theory - haven't yet actually completed a rug...) and I did an entertaining experiment with weaving-in-the-round: I used my daughter's hula hoop as my frame and tied T shirt strips across it to form a radial warp; I then wove long strips of stretchy T material around and around, expecting to generate a nice flat round rug. But when I cut it loose from the frame the tension of the strips released, and I ended up with a soft-sided bowl! Quite cute, but totally unexpected. Here is my cat asleep in the bowl:
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Sorry about the digression. I've never tried the scrappy thing, and I agree that you'd need to stabilize everything.
Alison