The last nominee without the support of a former president was probably Carter, since all of his Democratic predecessors were dead. Many Democrats between the Civil War and the retirement of Harry Truman did not have the endorsement of any former president, mostly because of the Republican dominance of the White House during that period (and the fairly short post-presidencies of Cleveland and Wilson).
In 1896, sitting Democratic president Grover Cleveland supported the "Gold Democrat" ticket of John Palmer over his own party's ticket of William Jennings Bryan. In 1912, Roosevelt was the lone living former president and supported his own third party bid over that of Taft.