Ha ha I have the same problem! I wash in stages, are use a bucket of hot water & and do it as fast as possible then cotton rug him.
Try cowboy magic spot remover, I have a couple of dirtbag horses at work that always get their legs or face dirty the morning of a show when there's no time to wash again. Just spray some on the stains and rub with a towel, smells very good too 😁 for the tail, we braid it up after washing and use a tail bag to prevent staining.
Maybe just use a cover cream on any areas that are really bad on the day and brush off when it's dry or use white chalk (or corn flour if you can't get any chalk in time hahah).
dry shampoo like veredus easy white. Just spray on dirty spot and wipe it off
Just curry/groom it really well. I like to bathe them and put on show sheen the night before, and put a nylon sheet on so they don't get dirty. Also polish the hoofs before entering the ring