FDR considered replacing Henry Wallace with Stassen in 1944 to drum up more bipartisan support for the war effort. Have this happen, and Roosevelt die on schedule on April 12, 1945, and BOOM! President Stassen.
However, you also probably have it impossible for Stassen to win re-election. Demobilization is going to kill his Presidency, as will the number of strikes. The Congress probably will go more Democratic in 1946 (with the public associating Stassen more with the Republicans than with FDR, as the public associated Andrew Johnson with the Democrats instead of Lincoln) and will hamstring the Stassen administration in every way possible with investigations and subpoenas, etc. One positive side effect (in my eyes anyway) is no Taft-Hartley, so the unions will probably be much more powerful down the line.
The Democrats nominate Dwight Eisenhower by acclamation on the first ballot of their convention in 1948. The Democratic platform for the first time endorses civil rights legislation for African-Americans, thanks to the prodding of Hubert Humphrey.
The Republicans are a lot less organized in 1948 than they were four years earlier. Defenders of President Stassen organize a bid to renominate him, though Senator Bob Taft leads him in the overall delegate count entering the convention. After many ballots and an exhausting procession of speakers, however, the convention renominates their standard bearer of 1944, Thomas Dewey, for the White House, denying President Stassen a shot at a second term.
Eisenhower drubs Dewey in the fall, and the Democrats continue in government for some time to come. President Stassen retires from politics for the time being and moves to Pennsylvania. He wins the Republican nomination for Governor of Pennsylvania in 1958 and loses in a tight race, but would go on to be elected Mayor of Philadelphia only a year later. Running a successful stint as Mayor, he received the Republican Gubernatorial nomination again in 1966, and would go on to be elected this time around, becoming the first former President elected to the Governorship of a state he had not previously governed.
Re-elected in 1970, Stassen publicly made comments about seeking the Presidency once again in the seventies, though it never quite panned out. Moving back to Minnesota following the end of his term, Stassen won the nomination for United States Senate of the Republican Party and would win election to the Senate that same year. In the Senate, Stassen defended his progressive Republicanism against the conservative shift of his party, though he decided not to seek another term in the Senate when his seat came up again in 1984.
He would, however, seek the Republican nomination for his local Congressional district in 1986, and would be elected to the House. Re-elected in 1988, 1990, and 1992, Stassen was drafted to hold the Republicans' senate seat by the Minnesota Republican Party in 1994, and would be elected to the Senate once again, serving with Democratic Senator Paul Wellstone. He would be re-elected to the spot in 2000, but died one year into his term in 2001. The sitting President of the United States, his old Senate partner, Paul Wellstone, would eulogize Senator Stassen at his funeral that year. He was succeeded in the Senate by Hubert Humphrey III.
Positions Held by Harold Stassen:
Dakota County, Minnesota District Attorney: 1931-1939
25th Governor of Minnesota: 1939-1943
Vice President of the United States: 1945
President of the United States: 1945-1949
Republican nominee for Governor of Pennsylvania: 1958
Mayor of Philadelphia: 1960-1967
Governor of Pennsylvania: 1967-1975
U.S. Senator from Minnesota: 1977-1985
Member of the U.S. House of Representatives, MN-4: 1987-1995
U.S. Senator from Minnesota: 1995-2001