Increasing the voltage beyond ~5.5V on my Samsung Galaxy S4 causes it to suspend charging.
On a friends phone, more than 6V stops charging and it brings up a message, "Charger overvoltage - Charging suspended"
A 5V supercapacitor will not charge the phone faster than a 5V USB connection. To charge it faster, you would have to access the battery terminals directly. You'd probably have to remove the battery altogether.