That chain appears to be designed to clip to the waistband of your trousers if you have a watch pocket. You will need a different chain for your waistcoats. One with a t-bar to put through a waistcoat buttonhole.

If it is long enough, you could put the clip in one pocket, thread the chain through the buttonhole, then attach to the watch and put it in the opposite pocket.

Nice watch, by the way.