I agree - we usually hide advanced search as well. However, looking at the templates for advanced search should allow you to see the form fields that control class filtering on search, and would provide some clues as to how to modify the regular search to do some of the same things.
I am not 100% sure, but if you saw something like this:
in the advanced search page, you could submit something similar as a hidden input on the standard search override. Or you could take entire components of the advanced search and put them in your standard search override.
