You may have pushed some of the existing dirt that was on the seat. Shouldn't be a problem to spread out the ring, so to speak. Use a damp microfiber towel with a light all purpose cleaner, like Simple Green diluted. Make sure to rub over the entire seating area with a little extra attention on the rings left from the snow.
If your seat was/is unusually dirty, it may be best to have a professional go over it with a hot water extractor.
John @ Hill City Auto Detail
Professional Swirl Remover!