How did people who disapprove of Lincoln vote?

[Sumner 1868]

1868-1924: Overwhelmingly Democratic
1928: Mostly Smith but sizably anti-Catholic defections
1932-1944: FDR, though less so every time
1948: Truman in the Upper South, Thurmond in the Deep South
1952: Stevenson without much enthusiasm
1956: " "
1960: A majority back Kennedy but sizable anti-Catholic defections
1964: Goldwater, unless they live within TVA range, Arkansas, Texas, or in Cajun country
1968: Wallace, with Nixon pockets
1972: Virtually all for Nixon
1976: Lean Ford, though just enough voted Carter combined with black vote he carried the South
1980: Reagan modestly
1984: Massively Reagan
1988: Massively Bush, by this point the GOP is the new Southern party
1992: Bush, unless they live within TVA range, Arkansas, or Cajun country
1996: Dole, unless they live within TVA range, Arkansas, or Cajun country
2000: W Bush easily
2004: W Bush massively
2008: Virtually all for McCain
2012: Virtually all for Romney
2016: Virtually all for Trump

[Sumner 1868]

Looking at county results, the idea that the Southern Whites who voted Ford had the majority of Lincoln haters is dubious at best.

Not really. Outside GA Carter carried most counties on the black vote plus a sizeable minority of whites. What you see on the map is the difference between thirty-something percent of whites voting Democratic vs teens or single-digits. Arkansas is an exception, of course, but that state always stood out a bit.