I would first make sure that you are dealing with Rain Rot, caused by a bacteria, and not Ring Worm, caused by a fungus. There are good descriptions of both at www.TheHorse.com
. It makes a difference as to how contagious it is, and how you treat it if it gets worse. Both microbes are common in the soil in many areas. Yes, you need to treat this, so that it doesn't get worse, and so that it doesn't infect other horses.
Iodine is a good general anti-microbial. Microtek is also a good shampoo with anti-microbial properties. I like and use the microtek; it's a little easier on the skin, and it won't stain you and your shoes. You need to leave both products on for 10 minutes, or so, to kill the microbes. Hopefully you have access to a heated wash room and warm water.
Because this happened after a move, it suggests that the horse is stressed by the move, and may have a compromised immune system. Remember, these are common microbes, and he is suddenly suseptable to them.
If the horse is blanketed, you will need to to wash the blanket in anti-microbial wash (maybe twice) and keep the blanket off of him the longest time possible each day. Wash all brushes, and don't share burshes between horses.