Try some rubberized ferrite sheet material - some are specifically designed to alleviate problems of this type (at 13.56 MHz). I seem to remember Ferroxcube making this product and I think Wurth do now: -
Wurth Elektronik WE-FSFS Flexible Sintered Ferrite Sheet WE-FSFS
adhesive flexible sintered ferrite sheets for EMI suppression and
magnetic shielding with a high permeability and low losses at
communication frequency. These WE-FSFS flexible ferrite sheets improve
the efficiency of Wireless Power Transfer (Qi-WPC, PMA, A4WP & WCT)
and improve the performance of NFC and RFID communication systems.
This also might be a useful read (again by Wurth):Going Wireless with Magnetic Shielding
I've used it to alleviate the energy losses in transferring 600 kHz magnetic power to a coil fixed to a stainless steel object. Without the ferrite behind the coils the power loss was just too great and I suspect this is what happens when you attach your tag to the case of your iPhone.