Hi Sven! It was fun talking to you last night.
IIUC this is the battery you have now… If you’re only looking to increase range, and you don’t mind a bulkier vehicle, you could just buy a second one and mount them in parallel… You might not be able to continue using the same charging system, but it would achieve your goal in a hurry, relatively safely.
But where’s the fun in that?
Last night I was suggesting a system involving cylindrical Lithium cells, the most popular of which are codenamed “18650B” mainly because of their dimensions (18mm diameter, 65mm length). I think for your use case you’d want a “4S2P” configuration, with four cells in series per bank, and two banks. Each bank would yield 12V when fully discharged to 16V fully charged, and having both banks in parallel would give you a good capacity/range boost with the same voltage.
@RolandClarke has a very similar power system for his amazing robot project, if you ask nicely he can probably tell you about the BMS (battery management system) he’s using, which takes care to balance-charge the cells to preserve their life and prevents over-charging and over-discharging.