An observation: there are horn teachers who don’t work with their students much on technical development. This is a shame. There really is a place for etude and scale study, multiple tonguing and low range exercises, etc., with students of all levels. The plain fact is there is underlying technique that must be built up to play anything on a high level, and to really play on a high level you must “over train” your technique to a very high level.
In auditions for honor bands and colleges scales are often requested. These should never sound bad! They must be “old friends” that you can toss off easily at a quick tempo. Shoot for 120 or faster in sixteenth notes. Without solid ability in scales you will never rise to a high level on horn. Or any other instrument for that matter.
In my teaching lately I have been drilling students to review the Clarke “Second Study,” an old classic. The bottom octave of this as printed in the popular trumpet technique book is an essential exercise. I especially like this exercise tongued and fairly loud, I feel this is very essential to return to as a part of keeping the chops healthy with good response in this critical mid-low register.
This register is somewhat neglected by players I feel as well in general. The sound quality above and below middle C must match, and it won’t match unless you work on it very aggressively for years. Besides the Clarke study mentioned above, there are many excellent mid-low range exercises in The Brass Gym.
One other suggestion: Mouthpiece choice in particular can greatly impact production in the mid-low register. If it is a problematic register, try something with a bit bigger inner diameter, it could do the trick!