It's an interesting question pitched when couched in those terms and one needs to remove the extremist views held ('die-hards' and 'eternal defeatists') considering any shifting Southern attitudes in the CW.
Thought it would be relatively late in the war, say by late '64, when the costs of conflict (destruction, deprivations and losses) were directly experienced by extensive areas of the Southern populace and the war really hit home. By this time, Sherman was practising his 'scorched earth' policy across the deep South in his 'march to the sea', and Sheridan had already destroyed the agricultural resources in the Shenandoah Valley while Lee's army was bogged down in a siege stalemate at Petersburg. In this same period, the South's two main fighting forces, the AoNV and the AoT, had suffered substantial losses of irredeemable manpower and were worn down by attrition. Additionally, the Confederacy's two remaining eastern seaports of Charleston and Wilmington, as sources of foreign supplies through blockade running for Confederate armies in the field were being strangled, and near capitulation.
Certainly by the start of '65, it's difficult to see in this dire situation how the vast general Southern civilian population (except for a tiny few dogmatic and stubborn politicians and military leaders) could have had any serious belief/hope of defeating the North or had any reason (or desire) to continue their suffering by prolonging the struggle.
(Perhaps a useful indicative exercise might be to examine a cross-section of Southern newspapers through the chronological period of the war and determine whether and when any of the bold headlines/captions viewed noticeably began to wane in hawkishness text/tones).