It has to do with angle of impact and the NW to SE direction of the small peninsula. As the hurricane passed over the peninsula going almost due north the first portion of the eye to regain water was the NE quadrant.
Hurricanes naturally seek fuel, and in the absence of sufficient fuel hurricanes in our hemisphere will tend to curve from the tropics to the nothwest through north and finally to the northeast. At that latitude hurricanes wihin the Gulf of Mexico have a natural tendency to the NNE. With the NE quadrant fiding fuel (warm water) first and the lifecycle tendency leaning that direction the hurricane naturally and predictably shifted slightly in that direction as it exited the coastal peninsula.
Yes, I know, sometimes hurricanes will move due west, even at that latitude and near land within the Gulf, but those cases are storms experiencing significant influence from adjacent pressure systems, and water temperature differentials. And in most cases those storms are already weak and continue weakening as they move counter to their natural tendency.
The short asnwer is that the minor shift was predictable. Speed, angle of impact, and strength at first strike are of huge importance when discussing the NO basin. I still fear that the destruction from this storm will prove to be massive, and that the early reports about 'dodging a bullett' are somewhat overstated. The Mississippi has yet to see the full impact of the rain runoff that will surely come in the following days. The potential for damage is far from over even though the winds may have passed.
They don't, actually. They're basically leaves floating in the stream of atmospheric winds, with the added "Beta Effect" of Coriolois which, all things being equal, continuously pushes them north.