I've just added another case to this list, which you can find
here. So far the total is 6 cases, three of them featuring a single clinic at Stanford (which won two of the three)! 6 cases is actually 7.6 % of the total of 79 "Opinions of the Court" that the Supreme Court lists on its
website. It's worth adding that my list so far is only of cases in which a clinic represented a party; there are probably many other cases in which clinics submitted amicus or friend-of-the-court briefs and these too can be influential.