My HAED is full of confetti, too. I work on 10x10 squares, using a sort of vague parking method. I use bigger needles (say size 22?) which I put vertically through the cloth underneath a finished square which has threads hanging (for example). Then I take those loose threads and wrap them as a bundle around the needle in a figure of eight. Keeps them neat. Others use other methods, but that works for me.
For an illustration, see Doreen's thread on HAED Spring Fairy piece.