He needs his hay increased and his grain decreased if at all possible. It sounds like he doesn't get to graze unless you are there, so he definitely needs an increase in roughage and probably a decrease in the amount of protein he is getting.
I usually see this kind of behavior in horses that aren't hyper, they are just unresponsive to the bit and are evading. I own an OTTB and when he first go off the track any pressure on the bars of his mouth would make him go. I thought he just had a lot of energy, so I lunged him before I got on at my trainer's suggestion. I ended up with a very sour, hot-headed and sore horse.
After getting my horse to a more relaxed setting barn and switching to a new trainer. My fast, nervous horse who only liked to run and me who was riding a half halt every step of the way, got called out by the new trainer. My horse had no mouth, he evaded the bit, wanted me to hang on his mouth because that's what he understood, and both of us were unhappy. My trainer, took off the elevator bit and replaced it with a french-link and took off the standing martingale. He explained that a lot of horses aren't properly trained to accept the bit so they learn habits to either go behind the bit creating a very choppy and uneven stride or they grab the bit and go...faster.
Don't let your horse fool you into thinking he is obedient because he stops when you ask him to in the small arena. So hopefully I can explain this correctly and feel free to ask any questions should you need clarification:
1. Next time you ride, if you feel comfortable doing so, don't lunge. Ride only in the smaller arena
2. Does your horse stand to be mounted? (make sure he does)
3. Ride your horse in a 10-15m circle. At the walk ask your horse for an inside bend by steadying your outside hand, increasing the pressure of your inside leg and opening your inside hand slightly.
4. The second you feel your horse slow down or drop his head, let the reins slide through your hands to encourage him stretch his head down. When you learn what this feels like, you will actually feel your horse get softer and his back will round a bit.
5. Pick your reins back up and repeat until you can get him to stretch his head down to the ground when you let the reins slide
6. This teaches your horse to reach out to your hands and lets him know that you will reward him by letting him stretch out when he responds to your hands and legs in a positive way
7. When you master this at the walk, do the same thing at the jog/trot and when your horse has this down, go to the canter. Remember the importance of riding your horse inside leg to outside hand.
8. Also, ride your horse in a spiral by opening your outside hand and using your inside leg to push him out laterally a step while he is still moving forward. This will make him slow down. So when you finally are ready to ride in the larger spaces, think "spiral" and he will slow down.
9. You can try riding half passes as well to get your horse to respond to your leg and also teach him to listen.
A half halt is not a slowing down aid as much as it is a preparatory and re-balancing aid. And a half halt is not the same for every horse. So if you are simply giving your horse a check in the mouth as a half halt, that's all it is...a check in the mouth. And if there is no reward ever, your horse has no option but to continue evading and going faster because he doesn't understand what you are asking. Your leg is as equally, if not more, important in the half halt as your hand. If you use a half halt correctly your horse will re-balance and prepare himself for what you are going to ask him which will usually cause him to slow down a bit.
"A quiet horse is not always obedient, but an obedient horse is always quiet."