How do you increase the torque of a DC motor?
You want to find the point where torque and RPM has the highest mechanical power and efficiency. Do this by measuring current vs. torque and speed vs torque.  

Using an adjustable torque load such as a small particle brake coupled to the motor shaft, increase the torque load to the motor just to the point where 
stall occurs. At stall, measure the torque from the brake and the motor current.

Construct a table of the motor mechanical power at various points from no-load to stall torque. Mechanical power output is the product of torque and speed. 



You can plot the measurements and calculated values





Using the Torque

On the robot the motor is geared down to a usable speed. You want the gear ratio set to give you the optimal mechanical power and efficiency where and when it is needed. 

You did not mention the criteria for winning or losing  making it difficult to go further.