zebra,

 First back down the strain screw, then hook the spring to the hammer stirrup and slide the back of the spring into the notch, once started you can tap it to the center. If the tension is to strong you can use a taped up jawed channel lock to get it started, I use leather to prevent scratches.

 I have no idea what a “coil spring update” is. I have never seen an original Winchester 1886 with a coil hammer spring. T/R