You can start new plants of Confederate jasmine from cuttings or from layering. Layering is the simplest method, though it may take longer. Simply lay a stem on the ground a bury a section of it, leaving the growing tip exposed. Usually within a few months new roots will form on the buried section, at which time you can cut the rooted stem off the mother plant and transplant it to a new location.
You can also start new plants from stem tip cuttings in June or July. Remove about 6 inches of stem tip, strip the lower leaves, dip the base in rooting hormone, and place the cutting in moist potting soil. Cover with plastic and keep moist until roots form, which may take several weeks or more.