It's kind of hard to be an "INO" president, since the president is generally very important in defining what the party stands for. The one president who really stands out to me is Carter, who had relatively little influence in Congress, was a bit to the right of the national Democratic Party, and who came reasonably close to losing re-nomination.
Going back further in history, Zachary Taylor and William Henry Harrison were WINOs, and Rutherford B. Hayes was a bit of a RINO.
John Tyler was even more so, and ironically the most Whig of all of them (Fillmore), ended up being seen as a traitor to a large segment of the party and helped to bring about its collapse. He also destroyed the American Know-Nothing Party, by splitting it on regional lines and driving its Northern wing into the Republican Party.