In the Samsung Galaxy S3 Stock ROM NFC does not work when screen is off or locked. With this mod NFC also works when screen is locked or off.

There are two versions. The first just enables NFC, when screen is on, but it also worked when it is locked. Also with a PIN or Password-Lock everything is fine. But it disables NFC when the SGS3 is in standby and the screen is off.
The second version let NFC active all the time, even when the SGS3 is in standby and the screen is off.
Nothing else is changed in the behaviour of the NFC.
I also created a ZIP to go back to stock NFC.

All ZIPs have to be flashed with CWM. You should wipe Dalvik cache after flashing.