How does rewinding a smart card antenna into a smaller coil leave it operational?
Without a link to the ring related page it's hard to be very specific BUT


There is much rubbish and many false claims published on the internet.This has a good chance of being less veracious than may appear. 
The system may still function but with reduced range.
It is exceedingly unlikely that it will work anything like as well as it did in its original form.


Patch antennas are based on a good grasp of the technology involved, a certain degree of magic and variable amounts of luck and experimentation. An antenna that worked in a card will have different area, different inductance and capacitances, different relationships of portions to other portions in terms of fractions of a wavelength and relative phase, and more. 

A look at Microchip's AN710 RFID Antenna design application note description page here  and PDF here will demonstrate the differences that varying geometry makes - and will also show you what may work best in a ring.