24v 450w DC motor powered electric bike connected to a lith-ion battery pack (56v 6.5a)
If 6.5A is the max discharge rate of the pack then the motor can only have about 15A put into it. At 24V that is only 360W, so if it needs 450W to do what you want you will need a different pack. 

Completely re-editing my answer after a string of useless comments. 

Not knowing what controller you're using makes this impossible to answer. The only suggestion I can leave you with not knowing the controller is us a DCDC buck converter to go from Battery-pack to 24V then feed that into the controller. As far as I can tell the only thing you changed that made the motor run extra hot was the input voltage to the controller. It used to be 24V then you made it 56V. Without knowing any other information, keep the input voltage to 24V and it should run similar to before. 

Another possibility though (and we can't know cause we have zero information about the motor also) is that it wants high current pulses and the battery pack can't do more than 6.5A. This would cause a rise in motor temperature. Good luck.