Wear any selected articles of clothing that you will need for the second costume concealed under your first costume, if possible. For example, if the first costume requires black tights and the second requires beige tights, wear the beige tights underneath the black. Keep in mind the excess clothing will cause your body temperature to be higher on stage, so you may not be able to wear two entire costumes at the same time.
Designate an area backstage or in the wings where you can change costumes without returning to the dressing room. Ensure the area is quickly and easily accessible but secluded enough to ensure privacy while changing.
Lay out the necessary pieces of the second costume in the order of the way you will need to put them on. Don't worry about tending to the removed clothes of the first costume, take them off as quickly as possible. The folding and hanging of these pieces can be tended to after the performance.
Ask someone to wait for you in the designated changing area to assist you with changing. This person will be vital when it comes to buttoning, unbuttoning, zipping and unzipping the two costumes as well as the pinning of any accessories. The hanging up and folding of the discarded costume can also be managed by this person while you are on stage.