Unless there's some specific reason you have to use JPEGs, I would recommend using PNGs instead, both because they have better quality and because they support transparency.
Also, if you really want true anti-aliasing on your cutouts, use Alpha Masking instead of the Magic Wand. To get the plugin, search for "alpha mask import" in the plugins section on this forum. To learn how to use it, check out the tutorials by Crazy Man Dan, both accessed from this page.