In my opinion, I would suggest getting to Site Manager before hopping to Site Director, as it does paint the image of hard work and dedication to be one step closer to that role; though your efforts and skills towards Site Administration, as well as departments you've overseen for certain periods will not be ignored. You are a competent figure for the role of Site Director, however Site Manager would suit you best at the moment.
Nonetheless, I wish you goodluck. Have my +Support.