Maybe.. but designing antennas is a difficult task; it is beyond me!
Without redesigning the loop antenna, you can maybe increase the range a bit by making a very good quality connection from the board to the loop. There are waves of electricity going through the loop, and if there is some obstacle for the wave to pass (like two wires joining), it can cause the same problem as an ocean wave trying to fit through a small gap: lots of ripples.
As far as I know, this circuit uses a combination of the loop, and capacitors, so that the waves in the loop antenna can resonate. Maybe you can think of this a bit like a guitar string: it needs to be the right length and tension so that you can get the right note.
This part of the manual talks about using solder to open or close J1 - J5. This can add or remove some capacitors, to try tune the "guitar" closer to your ideal "note".
Maybe it is possible that the module is able to deliver more power, and a bigger loop could help. Maybe the module can't deliver more power, but a bigger loop with less turns could possibly get a better reception than the smaller loop with lots of turns (?)
The difficult part is: if you try to make a new loop, it needs to have the same properties as the old loop, otherwise the "guitar string" won't make the right note anymore. The property to measure is "Inductance". I think it would be difficult to experiment with new loops unless you have some special tools: an LCR meter to measure the "Inductance", and an oscilloscope to measure the voltage of the wave in the loop.
Like I mentioned at the beginning of this post, I don't have much knowledge in this area, so don't listen too closely to any of this. I believe that there is a lot of very precise design and math that should really be used in this situation.
I did do a few rough experiments a few weeks ago, I'm not sure if there is anything helpful to see there or not: Automatic Dog Door wtih cam without collar - #31 by toddnz