Spray some odor-removing spray for fabric onto the surface of the coat. Be generous and spray the whole coat well. Hang the coat in a well-ventilated area or even outside to dry.
Apply a small amount of tea tree oil to a clean cloth and then wipe the cloth over the surface of the coat. (Tea tree oil is also really good for your skin, so you may want to keep some on hand to use as an astringent later.)
Fill a spray bottle with vinegar. Lightly spray the fabric of the coat, but don't go nuts because the smell of vinegar can be just as bad as that musty smell. Use just enough to lightly cover the surface of the coat and then hang the coat outside or in a really well ventilated area to dry.
Mix together some water and a little baking soda to create a liquid mixture. Dampen a cloth with the mixture and lightly blot the fabric on the coat to help absorb that musty smell. Don't use too much baking soda, though, or you'll end up with a white paste on the coat instead.
Place a container of open baking soda in the closet where the coat is hung or set a bowl of activated charcoal in the closet. Both will help absorb the smells and odors in the closet and help keep that coat and your other clothes smelling just fine.