Charging Supercap with lower source discharging with higher
It's not so much about being efficient as being impossible. If you have a 2.8V source, you can only charge the capacitor up to 2.8V. It will then discharge at 2.8V to start with but this voltage will gradually decrease as it gets discharged.

To get 5V out of it you will either need to use a step up converter to charge it to 5V, or feed it into a step up converter when you discharge it, or both. Using a converter on the input means you store more energy in the capacitor and hence you can get more out of it later. Using a step up converter on the output means you can maintain a steady 5V output despite the voltage gradually reducing during discharge. Doing both maximises the length of time you will be able to maintain that 5V output, if that is what you need to do.