You can really work on transitions. Before you canter get him to listen with walk trot transitions, and when you canter come back to a trot regularly. This should make him anticipate a transition and become more respectful
Also I agree with those below that more pulling isn't the answer, you have to make sure to give release for the smallest try in the right direction, that is how they learn what the right answer is. and it's great you are looking for answers besides a harsher bit. One of my favorite trainers always says "the bit doesn't control the horse, the horse has to control himself".
Why is he running through the bit? Is he anxious? (If so I'd work on focus and balance exercises on the ground to get that under control and you will find it helps his mental state when riding as well). Does he respond well to pressure at other times? I.e. at a stand still can you pick up on one rein with very light pressure and have him bend laterally, flexing at the poll without cheating and tilting his head too far sideways? Can you ask for a backup with light pressure on both reins and have him rock back and break at the poll as opposed to putting his head in the air or breaking at the third vertebrae and tucking his chin? Working on his mental state (is he paying attention, is he thinking or reacting, is he anxious?) and making sure he is responsive to the rein in general will both help this problem. If you can look at all those things honestly and they are good, then I would make running through the bit hard work by making him bend and do small circles, change direction quickly etc. One horse I help ride was very very rushy and out of control to the point the owner going to sell her (she was supposed to be a western pleasure horse but was out of control -rearing if you put pressure on the reins but running around like crazy). As soon as she started to rush the trainer I work under would bend her in small circles (using one rein only and with very light hands due to the rearing problem), and as soon as she slowed down the trainer would release and let her go straight again. She has a nice engaged western jog now and is much more responsive. The trainer actually rode her in a halter before going back to the bit because she was so used to being pulled around by the face her mouth was worn out so she needed to relearn what the right answer was when the rein was picked up.
Use trot poles up to 8 trot poles and align them so he has to condense his trot. if that doesn't work create a small circle path. Where if he tries to run out from you. it's only going to make his job more difficult.
If you ride western curb its may help. If you choose a right one they aren't severe and they give you more control wit less pulling
I first want to say that it's very good of you that you don't want to go to a harsher bit, because so many people think the bit is the answer. But I would just try to sit back and relax. Do a lot of half halts, and maybe even try taking him over trot poles to slow him down.
The main thing is half halting and staying light in your hand. It's a normal reaction to want to pull when the pull, but this will only make them harder in the mouth. When you are light with your hands they should eventually become lighter as well. Also remember not ho hold pressure, and to give when they listen. Also remember to reinforce your hand with your leg, seat, and voice.