Personally, I see a horse that really wants to move OUT and FORWARD that feels trapped by your hands and stiff riding, causing her to suck up and back, shorten her stride, and raise her head. Don't worry about her being hot. Let her find her own stride - a big, loose swinging gait where she steps under herself and really has to work her hind end. It'll feel giant to you, and probably a little overwhelming the first few times around the arena. Just tell yourself that you actually want her big, fast trot (which you probably consider her "hot" trot) - with time, she'll learn to slow down and carry herself and still move forward without the speed that likely causes you to pull on the reins and cause more frustration for her. The bit isn't the problem in the least. Now, I'm not saying you shouldn't have contact. You should ALWAYS have light, steady and giving contact. If she raises her head, you raise your hands and keep the same contact. She can't learn to brace against them. You just need more leg, and a lot of it. Everything else will follow, I promise! Once you have a steady rhythm and she is carrying herself consistently, you can begin working towards a stretch in the trot.
If you want to hear what I'm saying from someone who can explain it a lot better, check out arttoride on youtube. ArttoRide - YouTube.
Will Faerber is an amazing rider and trainer with a wealth of information that every rider can use, but especially those in a situation like yours. I think you'll see (and FEEL!) a difference from just one ride if you just give her more leg and don't get scared from the bigger, more swinging trot. I haven't mentioned cantering at all, because you can't achieve this at the canter until she's using herself to her full potential at the trot, and therefore there's no need to be working on faster gaits yet.