NTAG203 and RC522 module

I bought some of these tiny RFID tags from Adafruit and have been trying to get them to work with an RC522 module, using this library. So far, no luck. I am using the example sketch "DumpInfo" and have gotten it to work with the card and key fob thing that came with the reader, but the tags give me nothing, even with it physically touching the reader. The actual reader I purchased can be found here.

Any guidance? Or does anyone have recommendations for a small, ideally somewhat flexible rfid tag and reader that would work at a range of an inch, inch and a half?

In looking through the library readme, I'm thinking the RC522 may just not be strong enough.

Advice on a cheap (10-15 USD) nfc module that would work with these little tags would be very appreciated.