They wanted Cuba before, during, and after the Civil War. It was their own southward aimed version of Manifest Destiny. Also, the Confederates had a problem with soil exhaustion, which required new lands to cultivate. The Confederates also had their version of the White Man's Burden, which wanted to bring the 'blessings' of slavery to Latin America.
They wanted Cuba before the Civil War because they needed more slave states in the Union to balance the free states in the Senate. That's no longer an issue if the South has formed its own nation.
And what's your basis for believing that the South was interested in expanding into Cuba after the Civil War? My understanding is the exact opposite, that the South was generally hostile to overseas expansion in that period.
From "The Political Economy of Grand Strategy" by Kevin Narizny:
"Amply evidence exists that southerners were reluctant to pursue an active grand strategy in the periphery. Throughout 1865-1941, the South was far less likely than the Northeast to support treaties extending American influence in the periphery, including an 1884 agreement to give the United States exclusive rights to build a canal in Nicaragua (in contravention of the Anglo-American Clayton-Bulwer Treaty of 1850), the annexation of Hawaii, the annexation of the Philippines, the establishment of a de facto protectorate over Cuba via the Platt Amendment, and intervention in the Dominican Republic via the Roosevelt Corollary to the Monroe Doctrine."
(I would also add the proposed annexation of the Dominican Republic during the Grant Administration to that list, which also was opposed by Southerners.)
Narizny then goes on to point out how the South had little economic interest in expanding in the Caribbean since the export market for its goods in the Caribbean was negligible, since the sugar and tobacco growers in the South would vehemently oppose such expansion since they were already competing against Latin American producers, and since aggressive expansion into the Caribbean would risk damaging the South's relations with its primary trading partners in Europe (who OTL accounted for about 93% of the South's cotton exports up to the First World War.) Those reasons all seem valid to me, and indeed would be even more compelling for the Confederate States (who are going to be far more dependent on European good will than the southern states were after the Civil War in OTL.)