As you get involved in breeding, one thing that will make your life easier is having breeding affiliates. You can assign any other stable you'd like as your affiliate by going to the Breeding Affiliates page in the Breeding tab. You can have as many as you want, and the other stable does not need to approve it.
Affiliations are useful because in addition to your own stallions appearing in the hypomate dropdown, all of your breeding affiliates' stallions will appear there as well. This can cut down on the time needed to hypomate mares, cutting down on the tedious aspects of breeding.