Your lungs are like a balloon that expands to fill the cavity created by expanding your ribs and lowering your diaphram. (Lowering your diaphram may also make your stomach expand, that's okay.)
The best way to increase lung capacity (according to a vocal instructor I've had) is to take a deep breath, hold it, and then try to inhale a little more air, hold it, inhale a little more, etc, until you can't do it any more. the more you practice this, the better you'll be able to develop (use) your full lung capacity. Do this every day, and you should notice some improvement.
One other thing you might consider is the amount of air you use to play the horn. You usually don't need to "BLAAT" it all out at once. Controlling your output to what is needed will help you to have enough air to get through the phrasing.
I hope this helps.