Suppose that if I can find the right insertion/extraction tool I could move pin 5 to pin 6 and move it to the correct contact on the 24 pin connector.
However that inserts a failure point as the 24 pin connector has to connect to another connector before the motherboard.
Wiring in the original 10 pin connector soldering it to the wiring from the 24 pin connector may be my best bet as it will be a surefire connection method.
If the wire colors are to standard on that adapter it looks like I may have to swap a few more wires around on the adapter unless they just used two yellow wires for the red wires.