you can try a TT. My limited experience with such bits is that , yes, you may get more "brakes", but you often have less stearing ability. I guess that has to do with that this kind of shanked , 'snaffle" (broken moutpiece is a better term) , does't transmit lateral signals as clearly. It is best used for neck reining.
So, it can bring the horse to stop, but is not always so good for steering. And, if the hrose is running sideways, through the bit and all, you would want to be able to use lateral rein work, no?
I would learn how to disengage the horse's hind end, in the bit you have (a true snaffle), to stop him leaking out sideways, and then start him going forward again. Work at the slower gaits and get good obedience there before going into running.
And, as JDI said, make sure that the hrose is not running like this because he is trying to run away from pain from the bit or the saddle.