I agree with several of the other posters, most folks can get along great their entire riding careers in nothing more than a simple snaffle. The curb bits come into play when you are looking for more; more finesse, more responsiveness, etc. The whole idea behind curb bits (any bit with shanks) is to be able to work the horse on droopy reins, no constant contact of any form.
Anyway, back to the point. I don't see much reason to change the bits you've got now, even the tom thumb (even though I'm not a fan of them either LOL). If you get a nice leather strap to put on that snaffle, that will stop it from pulling through her mouth.
Now, I'll just mention this because I'm passionate about bits and their uses and how much a change in bit really can make a difference. If you wanted to try some different styles of bits on the mare that you're currently riding in a TT (or any of the others once they are solid with neck reining), here are some nice, mild, affordable bits that would be good options for your average rider and horse.
Most of these bits, I use bits that are either identical or very similar in mechanics and they are good bits in spite of the low prices.
Saddles Tack Horse Supplies - ChickSaddlery.com Reiner WIde Port Swivel Shank Bit