I fed my horses coconut oil as a supplement for a while. It made their manes and tails and coat so soft and their hair grew like crazy. It got expensive with two horses though. I had a big bulk jar of it and once that was done I stopped feeding it. It was an awesome experiment though and if I was showing I would probably feed my show horse the coconut oil again.
I think there is a couple of things you can try without spending money on supplements. First of all, be careful when brushing or combing, you don't want to rip the hair out. Start at the bottom and work your way to the top. Once all the tangles are out, brush from the base of the hair all the way through the tail which will move the natural oils down, it encourages growth of the hair as well as protects it. Then I would braid the tall for additional protection.
My mare rubbed HALF of her mane off last autumn! Now about a year later her mane is all the way down past her neck and I never cut it or anything, it's all the same length. So that short part grew out to the same length as the unrubbed part, just by brushing a few times a week moving the oils through the hair. That was when I knew for sure short hair really does grow faster. It didn't make any sense to me that the rubbed part of the mane would grow at a faster rate than the unrubbed half, but that is what happened! Kinda crazy. Anyway, her mane is gorgeous now and I have also noticed it has gotten thicker. That is also because I have my horses at home now and there is nowhere for her to stick her mane through a fence and rub it off.