DIY NFC boosting antenna for a mobile device
When I first read your question I assumed the following:


You are trying to track your clients' movements using your clients' smartphones 
Your clients would have to install an app on their smartphones in order to enable this


Then I saw the discussion about hacking the smartphones to change their antennas. This would imply that you would be providing the phones and giving them to the clients.

Hacking a cellphone to make it read NFC at a greater distance, would IMHO be quite a difficult task. As has been pointed out, NFC is specifically designed to limit distance (for privacy and security considerations, I believe).

Unless I'm missing something... if your scheme requires giving them something to carry with them in the clinic, you might just as well give them RFID tags, and place RFID readers in the doorways.

On the other hand, if you want to use the clients' cellphones with a downloaded app, as another poster suggested, Bluetooth low energy beacons might be a better approach, as BLE is designed to be readable at a greater distance.