I am battling something similar.

I have a cheap cig lighter voltmeter. Very handy...

At 1500 rpms and higher, I am getting 13.1 to 13.7 volts and it jumps around.
At idle, stopped at a stop light, I am getting 11.7 to 12.1 volts and it jumps around.

I was sure it was a voltage regulator and I replaced same with no change.
This alternator is a Bosch "Premium" rebuild of unknown age, but it is actually a Valeo 120amp alternator. It looked the voltage alternator had never been touched.

I have a new (not rebuilt) Valeo alternator on the way

OP- when you say you have a NEW Alternator in it, is it actually new or rebuilt?
I am hearing a lot of troubles with Bosch Rebuilds...which are no longer actually owned by Bosch btw...

The hot jump terminal is likely a big clue in your case....Maybe there is a short somewhere in the starter/alternator/jump point circuit.