Tennessee: Cordell Hull in 1945 (FDR's VP) Estes Kefauver in 1956, Al Gore Sr in 1963 (JFK's VP), Howard Baker in 1980, Al Gore in 2000, Bill Haslam in 2012
Get Jo Byrns instead of Haslam, and this might work. Basically have Jo Byrns live a few years longer. FDR picks Cordell Hull as his VP in 1940, then FDR dies in '41 and Hull dies not long after (or his perhaps killed). Speaker Byrnes is now president. Then Kefauver, Gore Sr., Baker, and Gore Jr. Maybe work Lamar Alexander in there somehow. Frank Clement might be able to work too but is probably mutually exclusive to one of the others.
That gives you seven, plus Jackson, Polk, and Johnson who are usually attributed to Tennessee, even though none were born in Tennessee.
If you want Haslam, the butterflies have hopefully changed him from OTL since OTL Haslam is considered pretty spineless when dealing with his own party/political by certain political insiders I've spoken with--not really presidential material, you might as well pick any other Tennessee governor.
I'd also say from what I've read, John Jay Hooker, Jr. might be presidential material in terms of Tennessee politicians. He was a Kennedy Democrat who I think if the main Nashville newspaper,
The Tennessean, had agreed to back him in 1974, he would've beaten Ray Blanton or any other Democratic candidate (and with that the election). They didn't OTL, since he'd lost the previous two elections, but it's still an idea. I think he'd keep the successful elements of Blanton's tenure and not create in the governor's office what some of Al Gore's associates called a "toxic waste dump". However, I don't know if John Jay Hooker ever had it in him to seek the presidential nomination. But successful economic policy could easily have been laid in the same period since the state had nowhere to go but up.
But let's say, for a wank.
*Andrew Jackson (D-TN, 1829 - 1837)
*James K. Polk (D-TN, 1845 - 1849)
*Andrew Johnson (D-TN, 1865 - 1869)
*Cordell Hull (D-TN, 1941)
*Jo Byrnes (D-TN, 1941 - 1945)
*Estes Kefauver (D-TN, 1952 - 1960)
*Al Gore, Sr. (D-TN, 1968 - 1972)
*Frank Clement (D-TN, 1976 - 1980)
*Howard Baker (R-TN, 1980 - 1988)
*John J. Hooker, Jr. (D-TN, 1992 - 2000)
*Al Gore, Jr. (D-TN, 2000 - 2004)
*Lamar Alexander (R-TN, 2004 - 2012)
That gives you twelve presidents associated with Tennessee.