RFID tag antenna pattern design

Hi friends,
I recently want to design a RFID tag antenna with working range about 10-15cm. The power supply is 3.7V to 3.8V.
I had designed a rectangular antenna with round corner, however, the working range was about 5cm which is to small to work. Therefore, I want to design an new antenna with another pattern in order to achieve the maximum working range with the same power supply stated before. Does anyone have some good ideas?

As I know, it is required omnidirectional antenna for RFID application.
I am not clear about your requirements in your application. Normally, in order to increase the Transmit range, two options are chosen, one is increase the TX power, another one is optimize the directive pattern of the TX ANT. For your case, double range is need to realized(5cm to 10cm), most easily, maybe placing reflection plane can work if you do not required omnidirectional pattern.

The requirement in the application is that the antenna can detect the tag in the same plane.
I thought the first way but it was little difficult to do on a PCB.
I am not very understand what is "optimize the directive pattern of the TX ANT", can have some explanations?
anyway, thanks you for your sharing