The Oracle of Attribution, David Michael Evans comes through once again. My thanks to him. I consider the case closed. This handling is superior to the one I was searching for! Here are the resultants of his research:
If the wallet is folded back to back, I associate that idea with Frank Chapman. In his method, however, there's no palming, and the wallet is carried in the inside coat pocket. This was published several times (with and without credit to Chapman). For example, starting with the original source...
Three Bits (Chapman, n.d.), pp. 5-6, "Again - The Card in the Wallet"
The Second Encyclopedia of Card Tricks (Gravatt, 1936), p. 67, "'Easy' Card in Wallet" - (uncredited)
Encyclopedia of Card Tricks (Hugard, 1937), p. 22-23, "The 'Easy' Card in Wallet" - (uncredited)
My Best (Thompson, 1945), pp. 42-43, "Again - The Card in the Wallet" - Franklin M. Chapman
Three Six Bits (Chapman, 1947), p. 7, "Again - The Card in the Wallet"
It might also be in an issue of Chap's Scrapbook, but I haven't checked.
THE "EASY" CARD IN WALLET
This method has several good points. The wallet is not prepared and the hand taking it from the pocket is empty.
Use a wallet that opens lengthwise and slip a heavy rubber band around one side. Open it so that the covers touch, back to back, and hang it over the edge of your inside coat pocket, the rubber band side in the pocket, the other side hanging out.
From any shuffled pack have a card freely selected, marked, returned, and bring it to the top. Place both hands with the pack behind your back. Take the marked card in your left hand, reach up under your coat at the back and push the card under your right armpit, retaining it with a slight pressure of the arm against the body. Bring the pack forward and throw out a card, any card, as you say, "Your card?" The answer will be "No."
Place the pack on the table, casually letting it be noted that your hands are empty. Take hold of the right edge of your coat with your right hand. With the left take the card from under your arm, slip it into the wallet, lift this from the pocket, flipping it over and closing it, and bring it out with the band side to the front. Put the right fingers under the band and pull it off as if it really encircled the wallet, open this and invite a spectator to take out the marked card.