Okay, I'm back, and armed with my full keyboard.
First off, the Tom Thumb bit is not a snaffle bit. The term "snaffle" is added on by manufacturers and retailers to trick people into thinking that it is a nice, kind bit. The fact of the matter is that it isn't a snaffle, and it isn't to be used like a snaffle.
Let's talk about the cheekpiece:
With a snaffle, the rein attaches to the same hole as the headstall - therefore when you pull with one pound of pressure, the horse is going to feel one pound of pressure.
With a leverage bit, you attach the headstall to one place, and the reins to a spot below the mouthpiece, and you have a curb chain attached under the headstall, near the mouthpiece. This completely changes the dynamics of the bit; when you pull with one pound of force, depending on the length and shape of the shank, the horse feels at least two. So, when you pull with 5 pounds of force, the horse feels at least ten.
Next let's look at what happens when you pull:
With a snaffle, when you pull the cheekpiece, you are translating exactly that pull to the horse's mouth.
When you pull the rein to a leverage (shank) bit, you are pulling below the bit, which twists the mouthpiece, and engages the curb chain. This means that the signal is not equal to what you expect pulling on the rein.
Here is an article I wrote about different cheek and mouthpieces:
http://www.horseforum.com/horse-tack-equipment/bit-information-snaffle-english-type-bits-36522/
When you are training a horse, you want to be very - VERY - clear about your communication. You want your rein to convey exactly that signal to the mouthpiece - thus you need a true snaffle bit. A curb bit should not be used to direct-rein, or keep contact with. A curb bit should only be used when the horse has learned to neck rein.
Here is an excellent article about the Tom Thumb bit:
Today's Horse - The Trouble with Tom Thumb
Please take a moment to consider this.
When you are mounting up, you are twisting the saddle, and putting lots of pressure on the horse's back; the pony might be a bit back sore. If the pony moves forwards, the usual response is to pull back on the reins... given what I've explained above, it is likely that the horse doesn't understand the extreme pressure on his mouth.
For your horse, I would recommend a full cheek french link bit; or, if you're worried about the cheeks getting stuck on anything, a Dee ring french link.