Path planning is often a high-dimensional computationally-expensive planning problem as it requires reasoning about the kinodynamic constraints of the robot and collisions of the robot with the environment. However, large regions of the environment are typically benign enough that a much faster low-dimensional planning combined with a local path following controller suffice. Planning with Adaptive Dimensionality that was recently developed makes use of this observation and iteratively constructs and searches a state-space consisting of mainly low-dimensional states. It only introduces regions of high-dimensional states into the state-space where they are necessary to ensure completeness and bounds on sub-optimality. However, due to its iterative nature, the approach relies on running a series of weighted A* searches. In this paper, we introduce and apply to Planning with Adaptive Dimensionality a simple but very effective incremental version of weighted A* that reuses its previously generated search tree if available. On the theoretical side, the new algorithm preserves guarantees on completeness and bounds on sub-optimality. On the experimental side, it speeds up 3D (x,y,heading) path planning with a full-body collision checking by up to a factor of 5. Our results also show that it tends to be much faster than applying alternative incremental graph search techniques such as D* to Planning with Adaptive Dimensionality.