Only way I could see it would be as a VP that takes over.
Then you have an far left president in an center right country. He leads teh dems to a mid term shellacking and gets crushed by a republicans when he runs for re-election.
At best a Carter.
And I don't really think his presidential campaigns really had much impact on the evolution of America's racial tolerance.
Considering social liberalism and social democracy are center-left, Bill Clinton is widely considered center-right, and Barack Obama is somewhere Right of a liberal and Left of Clinton, how does the implicit parallelism with Obama ("midterm shellacking") track? Do people get to make up their own political spectra now?
And, and I know I'm rusty, but wasn't Carter criticized for being to the economic Right (pre-Reagan fiscal conservatism) of the New Deal Coalition (liberals/social democrats; center-leftists)?
So the explicit Carter parallel doesn't track either.
And during the 1970's the U.S. was shifting to the center-right, as you said, though without a Democrat in the seat during the stagflation crisis there could have been hiccups with that; if we're planning a TL.
If people aren't able to pawn a right-of-center-left guy like Carter as leftist and to use a Democratic scapegoat to wipe off the shame of a corrupt Republican presidency (now there is a parallelism with today), then you might not end up with the Reagan presidency and so even with the neoliberal takeover of the Republican Party the nation might simply track straight center rather than go from center-left in 1960 to center-right by 1980.
Not that such would probably greatly improve Jackson's chances, but every little bit helps.