I dunno about moving your hands, but think about how the layers overlap. If you start at the middle and wrap to the outside, then the exposed edge of tape is towards the inside of the bars. As your hands press on the bars, particularly when pressing on the outer curve of the tops, you will 'uncurl' this edge.
Once, just once, I made the mistake of wrapping some bars this way. I thought it would give more grip. It was terrible.
Start at the bar ends and work to the inside.