I was in your shoes once and what I learned is that you have to shift before the hill. When I'm on group rides now, I see a lot of people doing what you're doing.. they wait until they hit that wall on the hill to downshift, but by then it's too late. Your gear combo may not be ideal for you climbing, and I would do what another poster suggested and try climbing in a higher gear once and a while and see how you do. As an example, on occasion I'll climb on the big ring and a smaller gear on the cassette.

Don't charge the hill in a high gear hoping to get momentum to carry you up and then shift down when you can't pedal. Instead, try and remain consistent when you ride. So many times, I see people sprinting on the straights and down hills and they may not realize it, but they're burning all of their hill climbing energy.. Then, they hit the hill.. and the wall and can't maintain any of their momentum, start panicking and downshifting.. but by then it's too late. Focus on a comfortable cadence and try to maintain that instead. I don't have a cadence meter.. but anything works, count on your own, count your pedal strokes as if it were musical notes. 1,2,3,4 - 1,2,3,4 and if you start to spin too fast or too slow, shift until you find that cadence.