How to create constant 5V 300mA from 0...6V variable voltage source (voltage generator)?
I think there might be a DC-DC converter for your on world's largest web store (mostly from China).
It costs about half the pack of cigarrets and just might be a suitable find for you. Try searching for: "1-6.5V to 5V buck boost" and you should find something useful.

In case you're not going to plug in any USB devices, just unsolder USB connector and find right wires to wire your application.

I hope it helps.

ps: I've used one when I did my solar charger for a phone. Works great!