If you want to change the horse's action, you need to change the cheek piece. My trainer says the mouth piece belongs to the horse and the cheek piece belongs to the rider.
Have you tried some kind of elevator bit? There's ones with three rings; you can ride in it like a pelham with two reins, so you only use the snaffle until you need a little extra, or some people just use the one rein. Metalab Jointed Three Ring Elevator French Link Bit, partrade.net
They're also nice because you have options, some horses, like mine, don't need much extra, so just the lifting action from the top two rings is sufficient, a boucher might also work in that case. ( Boucher Bits at Dover Saddlery
But if your horse needs a little more control, you can drop the reins down a hole or two, and hopefully get what you need!
Hope this helped :)