I agree with Thomas and his description of the actions your body is conducting at that time and the cues that go with it. They're all used to get your body under the bar. This needs to be a vigorous pull under, maybe just as vigorous (if not more) as your second pull.
I would add that you never want to pull with your back, either. The drive should be generated from actively pushing against the ground, as if you're trying to drive your feet through the ground. Granted, a strong core/trunk is essential for a balanced pull and receiving position, but the majority of the height you get from the bar is through your leg strength/extension and the power it generates into your hips as they extend as well. Don't simply "open" your hips; rather, simply focus on always pushing against that ground and the hips will have no choice but to extend fully.
As for movements to help, again, I agree with Thomas. In addition to the pull + clean, you might want to work on tall cleans to understand what we mean in regards to what your traps and arms should be doing. From there, work down the legs at different heights (mid-thigh, knees, below knees) in hang positions.
I hope this helps. Happy New Year!
Javier A. Sanjuan
Olympus Barbell Club
Dear God, please help me lift heavy and be awesome. Thanks. Amen.