The question of the relationship of former Presidents with incumbent Presidents is an interesting one, with usually the former Presidents avoiding open criticism of their successors, even if they are of a different political persuasions, and did not support the nomination or election of their successors.
There are only a few cases of open criticism and attack, including:
John Quincy Adams highly critical of Andrew Jackson, and returning to Washington, DC as a Congressman to “keep watch” over his policies and actions. Adams was also a sharp critic of the slavery and expansionist policies of John Tyler and James K. Polk.
Martin Van Buren being a major critic of the expansionist policies of John Tyler and James K. Polk in the 1840s, and of the slavery policies of Franklin Pierce and James Buchanan in the 1850s.
John Tyler, Millard Fillmore, Franklin Pierce, James Buchanan all critical of the policies of Abraham Lincoln during the Civil War.
Grover Cleveland being a sharp critic of Benjamin Harrison, who he had lost to, and then ran against again and defeated in 1892, and then opposed William McKinley and Theodore Roosevelt on the issue of imperialism and expansion.
Theodore Roosevelt very critical of his successor William Howard Taft, who he ran against on the Progressive Party line in 1912, and then against Woodrow Wilson’s policies toward World War I, after losing to him in 1912. Also, TR was resentful that Wilson “stole” some of his progressive ideas, and enacted them as President in his first term.
Herbert Hoover harshly critical of Franklin D. Roosevelt’s policies during the Great Depression and World War II.
Harry Truman very critical of Richard Nixon for years before he became President, and never really making peace even when Nixon gave the Truman library the piano in the White House that Truman had played. Also, Truman was critical of Dwight D. Eisenhower, and the two men only resolved their differences at the funeral of John F. Kennedy in 1963.
Jimmy Carter very critical of the policies of his successor, Ronald Reagan, and at times, of George H. W. Bush, Bill Clinton, George W. Bush, and Barack Obama.
Otherwise, the tendency has been to sit on the sidelines and avoid open criticism of one’s successors to the Oval Office!