Throughout the growing season, you can clip and use leaves as needed. Just cut back the stems, just not too much.
To harvest whole stems for drying, you can cut plants back to 4 inches above the soil line, just before the flowers open. Hang the stems to dry in a warm and dry place. If you grow a lot of hyssop, and are drying several sprigs at a time, wrap a rubber band around base of a bunch of hyssop, and hang from that rubber band upside down. I have a peg board in a dark room that is just off the garage where i do this. It works perfectly.