Wiring in new wire at tail lights

Without pictures, I cannot picture!!! what you are doing. If you are adapting/ splicing, they you are going to have to figure a way to .......adapt.........and splice. Consider heat shrink instead of "friction tape."

Soldering can be a chore on old wiring, as it oxidizes, even back inside the insulation. You must CAREFULLY scrape it GENTLY to get down to copper, and you should obtain some NON acidic (for electronics) liquid or paste rosin based flux to add additionally. And make CERTAIN that the solder is rosin core, FOR electronics. I would use an older lead/ tin mix, NOT "lead free"