Here's one thought on why a non-slaveholder Southern fought, not normally advanced.
Peer pressure certainly provided the motivation for many young men to volunteer.
Jacob R. Raper, Co. K, 43rd Tennessee, CSA, gave his reasons for joining up as:
“. . . the prevailing sentiment at that time in [my] district was that way, [his family was there and] the feeling at that time was quite strong against anyone who refused to go into the service.”
Also, the young ladies of the community let the boys know where their duty lay.
Here's Scotsman William Watson's take:
LIFE IN THE CONFEDERATE ARMY BEING THE Observations and Experiences of an Alien in the South During the AMERICAN CIVIL WAR
WILLIAM WATSON, New York: Scribner & Welford. 1888.
The Governor now called out the State Volunteers, and amongst others was the Baton Rouge Rifle Company, of which I was a member, although it was known to be nothing more than to make a military display, and on any other occasion would have been much enjoyed by the men. On this occasion, however, the duty was very repugnant to the feelings of most of them.
In our company the most of the men were strongly union in their sentiments, and as citizens were opposed to the whole
secession movement. Nevertheless, as they were in the service of the State, it was their duty to obey the orders of the Governor, who was by virtue of his office commander-in-chief of the State troops. They therefore responded to the call, although I for one confess, and I believe I might say the same for many others, that I would much rather have been called upon to act with the United States troops to suppress the secession movement and maintain the Union.
When Mr. Lincoln’s war proclamation was issued calling for troops to crush the seceded States into obedience, then in reality did unanimous secession take place. The people within the Confederate States, especially those who had been favourable to union and opposed to secession, were now face to face with the situation.
Whatever might be their sentiments, they could no longer remain passive. To submit to Lincoln’s terms they could not,
even if they were inclined. They were hopelessly in the power of the Confederate Government, which was strong, determined,
and unscrupulous.
If the secessionists fought and resisted Lincoln’s proclamation, the Union party could not remain neutral, they would be coerced and pressed into service, and compelled to take part in the resistance, or they would be persecuted or banished, their
goods and properties confiscated, and their homes desolated. They considered that the Federal Government by its weakness
and inaction had placed them in this predicament, and now the same Government, under a different leader, sought to chastise them for being in the position they were.
They did not long hesitate, however. They had sought to be loyal but received no support. They had been cast off and
abandoned, and they would not now submit to be whipped back into and under a Government which they considered had
forfeited all claim to their respect or allegiance.
The whole population within the Confederate States was now roused to a man, and the last vestige of Union sentiment was now cast off. Those who had hitherto been the most earnest in their desire to maintain the Union and preserve peace, were the first to take up arms. Nothing was now thought of but war, and resisting to the utmost the aggressive invasion threatened by Mr. Lincoln. Volunteer companies were raised in every community, armed and drilled with great perseverance.
I must again observe that whatever may be said or has been said on the abstract principles which led to this war, particularly where it has been set forth that the North was fighting to abolish slavery, and the South was fighting to maintain it; I must to such an assumption give an unqualified denial. Whatever may have been the question from which the quarrel originated, it had now been entirely departed from and lost sight of.
Of the thousands who at this time rushed to arms, I believe very few had in their minds the question of slavery. Many had before them the question of "States’ Rights" versus "Federal authority," but the greatest number were animated only by a determination to resist Lincoln’s proclamation.
THE storm of indignation which was aroused by Lincoln’s proclamation was not confined to the six States now forming the Confederacy. It spread like wildfire throughout all the other Southern States. The sympathy which the Confederate States
had lost by their rash act in bombarding Fort Sumter, was now recovered and increased tenfold.
I may here say that as the matter has reference to a volunteer movement, and a power, or nation, going to war or preparing to resist invasion with an army composed entirely of volunteers (on this subject I will hereafter speak), nothing could exceed the avidity and enthusiasm with which the cause was taken up. Every other object and consideration was set aside, the whole thought and conversation was centred in the one subject. Every man, woman, and child seemed to vie with each other in their efforts to aid and support it. Aged men and women furnished donations in money according to their circumstances. Poor families set to work in preparing shirts, underclothing, stockings, and other necessaries. Wealthy merchants and employers, whose employees and clerks would volunteer for service, made provision for their families or dependents by continuing their salaries during the time they volunteered for service.