The manufacturer is referring to a NXP Mifare chip, in other words, the implemented standard is ISO/IEC14443 A.
Of course, the only available option in your case is increasing the reader coil size. But the tag coil is obviously very small, so you won't achieve a large operation distance, even with a large reader coil. Besides field strength at the tag, receiving a sufficient load modulation signal is the second problem. As a first step, the required field strength and available tag signal, that are probably below ISO 14443 standard requirements, would be determined in a measurement.