The Other Celia by Theodore Sturgeon. 1957.
Appears in these:
https://www.isfdb.org/cgi-bin/title.cgi?45950
Story here:
http://www.baen.com/Chapters/9781625791061/9781625791061___2.htm
Slim heard more water-running and sudsing noises, and, by ear, followed the operation through a soaping and two rinses. Then she came out again, shaking out the object, which had apparently just been wrung, pulled it through a wooden clothes-hanger, arranged it creaselessly suspending from the crossbar of the hanger with the bar about at its waistline, and hung it with the others on the wardrobe door.
...
He folded it down flat again to be sure he could, and then brought more of it out. He soon realized that the material was of an irregular shape and almost certainly of one piece, so that folding it into a tight rectangle required care and great skill. Therefore he proceeded very slowly, stopping every now and then to fold it up again, and it took him more than an hour to get enough of it out so that he could identify it.
Identify? It was completely unlike anything he had ever seen before.
It was a human skin, done in some substance very like the real thing. The first fold, the one which had been revealed at first, was an area of the back, which was why it showed no features. One might liken it to a balloon, except that a deflated balloon is smaller in every dimension than an inflated one. As far as Slim could judge, this was life-sized— a little over five feet long and proportioned accordingly. The hair was peculiar, looking exactly like the real thing until flexed, and then revealing itself to be one piece.
It had Celia Sarton's face.
...
She scooped out the thing hidden there, shaking it once like a grocery clerk with a folded paper sack, so that the long limp thing straightened itself out. She arranged it carefully on the worn linoleum of the floor, arms down at the side, legs slightly apart, face up, neck straight. Then she lay down on the floor, too, head-to-head with the deflated thing. She reached up over her head, took hold of the collapsed image of herself about the region of the ears, and for a moment did some sort of manipulation of it against the top of her own head.
...
So it was that on the eighth day of his survey, a Tuesday, he went again to her room, got the bag, opened it, removed the box, opened it, removed the ream of paper, slid the blue band off, removed the covering sheets, took out the second Celia Sarton, put her on the bed and then replaced paper, blue band, box-cover, box, and bag as he had found them. He put the folded thing under his shirt and went out, carefully locking the door behind him in his special way, and went upstairs to his room. He put his prize under the four clean shirts in his bottom drawer and sat down to await Celia Sarton's homecoming.
