Going off the last map of the 1996 Senate race, I combined the Republican and Democratic votes in the primary and then compared that to the runoff. Landrieu was very lucky that the runoff was held concurrently with the Presidential election (back then, the primaries were in September or October).
In the 1996 primary, Republicans candidates collectively got 56% of the two-party vote. Comparing the partisan numbers primary to the runoff might not be exactly apples to apples, as the jungle primary emphasizes candidates over parties. Still, Landrieu was lucky to be on the ballot with Clinton for the runoff.
Here's the breakdown by parish. The first map is the two-party primary vote, then the actual Landrieu/Jenkins runoff, then the swing:
61 out of the 64 parishes got more Democratic. Calcasieu (Lake Charles) and Cameron Parishes swung Republican. The Democrat that Landrieu narrowly beat in the primary, Richard Ieyoub was from Lake Charles, so his home-region effect didn't translate over to Landrieu. Ieyoub was very popular there; he served as Calcasieu Parish DA and then went on to serve as the AG.
West Baton Rouge Parish also swung slightly R. My guess is that there are a lot working-class whites there that would have reacted better to Ieyoub than Landrieu. Ieyoub was very much in the tradition of the Longs, while Landrieu was seen as more of a modern Democrat. Jenkins was (is) also from around Baton Rouge, so that may have helped.
Next, here's the breakdown by CD. Again, the first map is the primary two-party vote. Though this was 1996, this map turned out to be a good preview of how the CDs would go on to vote in the 2000's. For instance, other than CD1, CD5 was the first district to give a Republican over 60% (Bush in 2004).
The second map is the runoff and then the swing.
Finally, I did looked at the turnout. Overall, the turnout jump from the September primary to the November runoff was 38.4%, or about 500K votes. Higher turnout in urban parishes helped Landrieu. Still, there wasn't really an overwhelming correlation between increased turnout and partisan swing. For example, the turnout spiked around Shreveport but CD4 only swung 10% to Landrieu (less than the statewide swing). Also, while the parishes around Alexandria all swung markedly D, their turnout was fairly static.
Caldwell Parish was the only parish to cast less votes in the runoff. Its a very small, rural parish (population 10K), so that's likely why. David Duke always did well there (even carrying it in the primary), so perhaps election enthusiasm there decreased with him off the ballot.