For several days I tried all possible and impossible things.
Conclusion:
With me fit S4 and Cover apparently not 100% together or two of the contacts, which have to divert the NFC antenna not connected. For me, it helped to put a small piece of paper on the contacts. If I then carefully cover the sit up, so that the paper I did not slip at once full NFC functionality.
Important:
If you flipped the S4 before has - top left is the battery contacts. Since Do not fiddle around. At the top right are the contacts for the NFC antenna. I'll try this to thicken with a little touch paint. Vl. then have the contacts even without paper a connection.