From the research I have done, the Schoonover surname didn't evolve until around 1800 (give or take). Before that, the Schoonover ancestors were "Schoonhoven." All but a few of the church records I've seen from Fort Orange, Kingston, and Minisink where the family first lived lists them as Schoonhoven. I honestly would have expected them to be written as "Van Schoonhoven" which is how most of the early generations are listed on the very robust Schoonover Mastertree.
In the spirit of naming people how they referred to themselves, I actually think the early Schoonovers of colonial NY should be renamed to "Schoonhoven" and the generations that changed should have that reflected between their 'Last Name at Birth' and 'Current Last Name' fields.
It is worth saying too that I have seen the names Schoonhoven, Van Schoonhoven, and Schoonover, but never Van Schoonover.