The setup is exactly the same right up to the point where I have you stretch the spun coil of wire. Instead, leave the wire tightly coiled and cut off your rings leaving no gaps at all.
This now leaves rings that are just barely twisted. It will take some practice to avoid nicking each subsequent ring with the tip of the wire cutters while you are working.
To connect the rings together pick them up using two pairs of needle nose wire cutters. You will have to invert one of them for ease in bending (this will also take some getting used to).
Open the link slightly (just enough to link in to another ring) and knit it into whatever mail project you are working on. It should be noted that this method is easier than the first method when working with exceptionally small links (e.g. 16 awg, 3/16" rings). Some people have reported having some luck with a hybrid of the two methods. In time, any armourer will have perfected slight (or not-so-slight) variations on these methods as their own method.