Most people don't think of this one......
My tip CLEAN TEETH. I used to volunteer at our stable for the monthly shows and time after time I saw good horses get passed over because although they were perfect in every way, they had green gunky teeth. Not that every judge likes to look over all the horses in the mouth but in Halter as well as even Hunters Under Saddle, I have seen some judges go to the mouth when they just couldn't decide.
My suggestion, before going in the ring, just take a quick peek and maybe have someone take a damp toothbrush and gently brush out those carrot shreds, horse cookies and alfalfa twigs that may jump out to cause you to receive a red or white instead of a blue.
Other places I have seen indecisive judges go....under the tail, in the ear, and asking the horse to pick up it's feet.