I disagree - here are a few examples (pulled up from memory):
By this Constitution, certain duties were imposed upon the several States, and the exercise of certain of their powers was restrained, which necessarily implied their continued existence as sovereign States. But to remove all doubt, an amendment was added, which declared that the powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States, respectively, or to the people.
South Carolina - Declaration of the Immediate Causes Which Induce and Justify the Secession of South Carolina from the Federal Union, 1860
---
Senators, the Constitution is a compact. It contains all our obligations and the duties of the federal government… I say that the Constitution is the whole compact. All the obligations, all the chains that fetter the limbs of my people, are nominated in the bond, and they wisely excluded any conclusion against them, by declaring that “the powers not granted by the Constitution to the United States, or forbidden by it to the States, belonged to the States respectively or the people.”
Robert Augustus Toombs of Georgia – Remarks upon resigning from the US Senate, 1861
---
The Constitution of 1787, having, however, omitted the clause already recited from the Articles of Confederation, which provided in explicit terms that each State retained its sovereignty and independence, some alarm was felt in the States, when invited to ratify the Constitution, lest this omission should be construed into an abandonment of their cherished principle, and they refused to be satisfied until amendments were added to the Constitution placing beyond any pretense of doubt the reservation by the States of all their sovereign rights and powers not expressly delegated to the United States by the Constitution. Strange, indeed, must it appear to the impartial observer, but it is none the less true that all these carefully worded clauses proved unavailing to prevent the rise and growth in the Northern States of a political school which has persistently claimed that the government thus formed was not a compact between States, but was in effect a national government, set up above and over the States.
Jefferson Davis - Message to Congress, April 29, 1861
---
Perhaps you were doing a quick search, using terms such as "Tenth Amendment" or "10th Amendment". What was commonly cited was the language (or paraphrased language) of the amendment itself...
No I read each of the eleven Ordnance of secession, and 5 documents, by the secession conventions, justifying their secession. I do stand corrected on the Carolina justification, it is the only one of 5 justifications that allude to the 10th amendment. The 10th is not cited in any of 11 ordinances of secession or in the supporting is the 10th amendment by the four states that wrote them.