There are quite a few moving parts to this question. You're right that, to an extent, propaganda was needed not only to convince soldiers to fight, but to convince the South in general to secede in the first place (and even then, significant Unionist minorities existed in every Confederate state). However, I doubt that propaganda could ever sustain the Confederate war effort (or any war effort for that matter). Soldiers, leaders, and partisans in the Civil War, Union and Confederate, were responding to deeply ingrained political, cultural, and economic viewpoints that had been molding sectional tensions for decades before the war began. If you put all this together, I hope you'll see that even a Confederate soldier from a lower class background, and who didn't own slaves, could still very much have a stake in the fight.

Lincoln's election in 1860 was a critical moment that a small but influential group of radical Southern politicians, known as fire-eaters, seized upon. The election, in and of itself, stirred up a lot of anger and resentment across the South but it did not make war inevitable. Fire-eaters capitalized on the shock of the election to claim that all the South's fears about the Republican Party and the North in general--that they were radicalizing, obsessed with abolishing slavery completely in the US, that a Lincoln administration would specifically target the South and seek to render it subordinate to the North. They pushed for immediate action on the question of secession so that several states left the Union quickly rather than get bogged down in debates over how to leave the Union, if they left at all. This isn't to suggest that the southern populace was pushed into secession against its will, but rather that the fire-eaters had been looking for their chance, found it, and recognized the importance of momentum.

But why would the North's apparent threat to southern slavery matter to a poor southern white who didn't own slaves? To understand that you have to understand how the South had developed economically, racially, and culturally for the previous half-century (longer, actually). The entire cultural and economic makeup of the South was based on slavery. Southerners' perceptions of distinctly American values (like the concept of 'liberty') incorporated slavery and racial hierarchy. Slave labor was seen as a significant uniting factor for white men in that it (supposedly) freed them from class and political division, as it prevented white men from having to work for other white men. Whiteness bestowed political, legal, economic, and social privileges that depended on the racial hierarchy that slavery guaranteed. Further, slavery was seen as the primary means of economic mobility for Southerners. If you're a poor white man and don't want to be poor forever, the acquisition of land and slaves to work on it was the primary path to wealth and prosperity. Denying either of those things, as the North was doing whenever it opposed slavery (especially the expansion of slavery to new lands in the West), was tantamount to denying Southerners a path to economic independence and thus true liberty. Slavery was a primary source of social standing, domestic authority, political influence, and economic independence in the antebellum South. Protecting it wasn't unlike protecting the essence of what it was to be an American (as they saw it).

So how was this used to keep poor Confederate soldiers fighting? Well, for starters, sometimes nothing worked. Desertion was a huge problem on both sides, but especially for the Confederacy (as most fighting took place in the South). There were consistent complaints about the Confederate side of the war being a "rich man's war and a poor man's fight," as well as plenty of resentment over conscription, especially the rule that exempted a white man from service if he had a certain number of slaves to oversee at home.

Aside from that stuff, propaganda around the Confederate cause mainly built on the arguments brought up during secession: that the North was a tyrannical invader bent on subjugating the South (an especially effective argument in a culture which emphasized masculine independence and authority), and that slavery would be abolished, and that their homes and entire way of life were at stake. It's worth noting that while these arguments are not wholly true, they're not wholly untrue either.

Finally, I want to bring up the idea of Confederate nationalism, which seems strange when describing an (attempted) nation so predicated on state sovereignty. As the war escalated, the Confederate government had to adopt centralized structures far beyond what one might expect, as well as cultivate a sense of national unity that would focus their cause. The primary national symbol that emerged was Robert E. Lee and the Army of Northern Virginia, and in many ways its successes and failures were seen as representing the nation as a whole. A significant willingness to defend this new Confederate nation (and not just one's respective home state) crept up by the war's end and only really dissipated when Lee's army itself disbanded.

Readings:

McPherson, Battle Cry of Freedom

Freehling, Road to Disunion (esp. Volume 2

Gallagher, The Confederate War

McCurry, Masters of Small Worlds

Oakes, The Ruling Race