You probably can't change the power output of this device significantly without significant modifications. It seems likely that significant modifications would yeild a device that is illegal in many countries.
You could add an amplifier to the output but as with "significant modifications" it's likely to be non-compliant.
Things that you might do is try to reduce the incoming signal strength of the competing radio stations - by removing/reducing the antenna to the receiver and placing the device closer to the reduced antenna. Clearly you'd need to install a switch for this to be practical but experiment first.
Adding more antenna to your device might work but it's hard to say. More wire changes the apparent load on the transmitter possibly negating any expected improvements.
To be effective, the improvements have to be profound. You won't notice a 20% improvement, it would have to be a 200% improvement or more.