I have climbed very few tough hills, but on the ones I climbed the best I reserved the lowest gears for when the climb got really tough, and each time I shifted down I felt relief in having another gear, which increased my mental (and physical) momentum. That is a trick I hope to remember on the tough hills in my upcoming metric.
I would say your strategy is backwards. I think you should shift to an easy gear that you can spin at a good cadence very early on the hill. You'll need to work at spinning to have this technique feel right. If you can ride at 85-90 rpm at the bottom of the climb it will save your muscles for the harder effort if the hill steepens. You might also have a gear left that you can drop down to to maintain the cadence. If not, you will be no worse off than if you pushed a bigger gear at a lower cadence and dropped down as you climbed.
As you get near the top of the climb, shift to a harder gear and use those muscles you saved earlier in the climb. Proceed to pass your riding buddies and show off your climbing skills.