How to increase the read range of an active NFC tag?
You can't.

NFC limits it's range with two contradicting requirements:


RF field from the reader has to be powerful enough to power the NFC tag.
RF field has to be weak enough that the NFC tag can signal data by consuming power (load modulation).


To increase the range you have to increase the RF field to keep the tag powered, but if you do that you make it harder for the tag to answer. 

That's why NFC only works up to 15cm. The range limit is designed into the technology and it's a security feature of NFC. If you want longer range look into a different RF technology.

There is a way to boost the range up to about 30cm by adding an extra circuit that amplifies the load modulation but this requires a large antenna and only works so-so because you don't only boost the answers from the tag but noise as well.