just pull the sheathing back and splice in the wire to the stock wire. no need to cut it completely.
makes sure, remove the sheathing, split the factory wire into about half, so you can slide the new wire through the middle of it, then wrap it around itself on the stock wire. then solder over the join. best way to bond the two together. And when doing the connection that is completely cut in half, fan out the strands on all the ends, touc the two ends together, messing the fan pattern, and then twist them all together and then solder.