I'll let others recommend wire gauges.
But - thanks for your other HO thread reply!
And, there are 2 issues:
1 - will your existing cable
safely handle the possible current?
2 - is the voltage drop for "safe" cable too high for what you want?
Instead of OFC you can use a larger cable if that is cheaper and cost is an issue.
From a design POV, it's the acceptable voltage drop from the alternator to the load. That's V=IR (Ohm's Law) where V is the voltage drop (volts), I is the current (Amps) and R is the resistance of the cable.
Note that R includes both the +12V and the return/GND path.
And R is
simply(?? LOL?)
resistance per length of cable times the
cable length - ie, Ohms per km times km. (Or whatever units, but make sure the Ohms-per unit matches the vehicle's cable length, ie divided by 1,000 for km to meters etc.)
But if you have only one battery and it's in the rear, chances are you have a very long and fat starter cable to the starter motor.
That cable should be used for the alternator to batt +12V.
FYI - Battery relocation to the rear usually means increasing starter cable gauge to compensate for the longer battery to starter distance, and the original alternator and/or fuse connections to the battery are reconnected to the starter motor. If the alternator has a Sense wire, that is run direct to battery +12V if voltage drops from the battery to alternator are an issue. For single wire D+ alternators, this is not possible (without alternator mods).
Generally when batteries are moved for audio purposes, people instead add a second "audio" battery and a battery isolator (UIBI or charge controlled, or voltage controlled - forget diode isolators), hence keeping the original cranking battery up front and adding typically an AGM battery in the rear (deep cycle else cranking type depending on what is desired from the audio system).
They hence get maximum life from both batteries as well as independence - ie, they can crank their vehicle even if the audio battery is flat.
As to the GND path resistance, the worst case usually assumes the same cable as the +12V path.
However in practice, the body/chassis is usually used which is usually far less resistance than cables - but beware of some minimal spot weld joins or non-metal sections - and hence GND cables are relatively short.
A simple doubling (duplication) of the GND cabling halves the resistance of that cabling. Same for the +12V cable, but if the GND cables are short....