As promised, below are my two OLED screens:
This is the where I got the 6-pin one:
https://www.ebay.co.uk/itm/128X64-0-96- ... 2748.l2649
And the 4-pin - note, that it was adertised as a 1.3" screen:
https://www.ebay.co.uk/itm/1-3-OLED-LCD ... 2749.l2649
Here are the circuit board side of the OLED screens:

And I tried to trace out the 4-pin one:

Now that I have got the 2.42" screen working, I will go back go to try to get the 6-pin screen working, as it has similar pins.
For the 4-pin one, I thought it might be just a drop in replacement for the working 4-pin 0.91" 128x32 screen (albeit the VCC and GND pins swapped, and no RES pin). However this didn't seem to work. I suspected Pin 20, which seems to be not connected. This is assume this is the D2 pin, and on some OLED screen schematics I've seen is connected to GND. I tried this to solder a jumper wire to it and connect it to a GND pin, but that did not seem to work.
I also thought that the 10K pull-up resistors on SCL and SDA might be issue (as some sources suggests that the pull up resistors should be 4.7K) so I stacked a 10K resistor on them to make then 5K. But this still didn't work (screen is just blank).