The way I splice wires is to strip about 1/2" off the end, twist them together end to end, so it looks like a continuous length of wire, then solder and heat shrink. Although time consuming, I believe this is the best way for reliability and electrical conductivity. For larger wires, a properly crimped butt splice sealed up with heat shrink is better.
OK, I read your question again. For tapping into an existing wire, I strip about 1/2" of insulation, without cutting the wire. The splicing wire is also stripped at the end and twisted around the exposed part of the other wire. Then solder and heat shrink.
Jim