I don't understand why you are "exporting a transparent image". After you extract the subject "jump" (
PS New layer via copy) to a new layer. In Photoshop just drag that layer with the transparency to a new document or to the other file where you have extracted the second subject.
The white border you got around your extraction was probably transition or anti-alias pixels around the hair after using the magic wand. Most folks would select the white background, invert the selection, modify the selection by contractng a couple pixels, set a small feather then copy the selection to a new layer (or invert the selection again so that the background is the selection then delete the background.)