Schools, parks, statues, public buildings, highways, that sort of thing?
There’s a lincoln county in Mississippi. Dunno if its named after the President or not, though.
Also, I believe there was something of a tradition of naming public schools for African Americans after Lincoln. At least some of these schools are still around (though presumably no longer segregated).
Not quite what you’re looking for, but there are monuments to the Lincoln family in Virginia, where they lived before moving west:
http://www.hmdb.org/marker.asp?marker=15634
I don’t know if you consider it “The Confederate South”, but this one is clearly a Memorial to Abe Lincoln in Missouri:
And there’s this statue to Lincoln in Richmond, commemorating a visit he made shortly after the city fell:
Here’s a statue of Lincoln with his dog (!) from Harrogate, Tennessee:
http://www.byways.org/explore/byways/76017/photos.html?id=78970
Here’s an entire Licoln Museum in North Carolina:
Statues of Lincoln and Jefferson Davis at Vicksburg, Mississippi:
http://www.nps.gov/vick/historyculture/kentucky-memorial.htm
Car dealerships honor his birthday annually, even in the South.
Southerners will make a monument to just about anything. Like say this:
Plenty of schools Texas. I went to Lincoln Junior High School as a kid.
There’s huge bust of Lincoln in downtown Houston alongside I-45 (North Freeway), alongside Sam Houston, Stephen F. Austin, and George Washington. Sculptor David Adickes humorously calls it Mount Rush Hour. (closer picture).
Fairfax County, VA has a neighborhood/census designated place called Lincolnia.
Even among diehard “The South Will Rise Again” people I’ve never detected any particular animosity towards Abe.
Oh, it’s there, in some cases. Certain families still refer to him, when they cannot avoid it, as “Mister Lincoln,” avoiding the honorific “President.”
Author Shelby Foote said in Ken Burn’s documentary that he met with a descendant of Confederate cavalry hero and KKK co-founder Nathan Bedford Forrest. She admired his scolarship, and during his visit, invited him to touch Forrest’s actual cavalry saber.
With great reverence, he drew the sword and swung it once overhead. He thehn confided to the lady that, in his opinion, the Civil War had produiced two authentic geniuses: her ancestor Forrest, and Abraham Lincoln.
She immediately took the sword back, saying stiffly, “We don’t much cotton to MISTER Lincoln.” Foote ruefully reported he never got to touch the sword again.
According to here, it is.
Texas is not the South, it’s the southwest.
Perhaps, but I take the OP’s “Confederate South” to be asking about former states of the Confederacy. Texas undeniably fits that bill.
It was very much part of the Confederacy, though, and I’d include it in the OP’s “Confederate south”. Much more so than Missouri, which was a border state and remained indecisive.
(Aside - William Least Heat-Moon, the author of “Blue Highways”, is from Missouri and commented how everybody in all parts of the country would categorize the state as from another region than theirs. It is rather indeterminently placed.)
You’re fooling yourself. 
There’s Lincoln Memorial University in Harrogate, TN…
It’s both. Or rather, it’s got both.