ESP8266 - does it really have terrible wifi range by default?
The very first thing I notice is that your WiFi antenna is shielded by the PCB. That is the antenna is placed right over the top of the ground plane of the PCB and your range is going to be severely limited.

You will notice on the NodeMCU design below that they have placed a PCB cutout where the antenna is located. In other pictures you can also see that the ground plane does not go past where the antenna starts.



On other boards the antenna sits proud of the main PCB such as this one from Tronixlabs:



If money is tight and new hardware is out, you could try cutting a cutout where the antenna sits using your Dremel (or other rotary tool). If there are any traces underneath the antenna you will need to fix them up with wire worms. Of course it is best to remove the ESP module first.

Alternatively, you could redesign the PCB and send it out for another manufacturing run. I use PCBShopper.com to find the best price for the bare PCBs.

Edit:
Something else you could do is to unsolder the ESP and mount it 90 degrees to the PCB. You would need to solder wires from the base PCB to the terminals of the ESP. This should work and you won't need to cut the base PCB.