There is a difference between a horse that snaps his knees up and a horse that jumps round with a nice bascule. Getting a horse to snap his knees up isn't usually too difficult with some nice gymnastic work. Getting a horse to round is harder to do as that takes more scope and natural ability on the horse's part, but can be done. Getting a horse to bascule AND snap his knees up? Take a combination of good training, good riding, and natural ability on the horse's part.
I actually just did a clinic with a grand prix trainer where we worked mainly on getting our horse's jump better. We normally do quite a bit of gymnastics as is but this weekend we did whole courses of them and the difference was unbelievable. My friend's horse who shows in the 4 ft and tends to drag poles left and right with his hind end was firing over the fence and kicking up his hind end by the end of the 4 days! At 4"3! We rode several horses every day and by the time we were finished, every single one of them was jumping better.
1) gymnastics! Lots and lots of them! I know several grand prix trainers who do nothing but good flatwork and gymnastics until they get to the show ring. It's the absolute best way to teach your horse how to jump and what to do wtih their legs and body. AND it teaches you as a rider to be still.
2) BEING STILL AS A RIDER! I don't know if you're a 'picker' but if you go down the lines picking at your horse all over the place they never learn. And if you go around a course being busy/unbalanced with your body they'll never use their body's well. This is crucial. There's a "big time" trainer in our area who buys these incredible jumpers from overseas. She goes into the ring picking and fidgeting and throwing her body all over the place. Every ride her horses get worse and worse and before you know it they enter the ring and knock down every rail. It's so ridiculous. What a waste. You have to let your horse use it's body without you interfering.
3) FLATWORK! To properly use his body your horse must be limber, agile, strong, and balanced. He must know how to power off his hind end, carry you to the fence, have impulsion before and after the fence, You acheive all those things with proper flatwork. Leg yeilds, shoulder-in, haunches in, half pass, renvers, travers, etc. Being balanced through serpentines, figure 8s, etc. We did almost a solid hour of everything I just listed BEFORE we started to jump. And again, all of our horses were better by the end of the clinic. Tired, but better.
Don't forget cavelettis. Those things are wonderful at getting your horse more coordinated, limber, strong, etc.
And just a thought... While yes, most horses do use their body more over the bigger stuff make sure that you're ready to do it first. I have no idea where you are in your riding but if you hit your horse in the mouth at 3"6 because you aren't capable of doing it securely then you're teaching your horse to jump flat and defensively.
Good luck! Sounds like your trainer is doing a great job!