One of the most important things to understand about the US dollar in the Confederacy during the American Civil War, is that official US paper currency as we now think of it did not exist prior to the war. Save temporary measures during the crises of the Revolution and the War of 1812, the US government had only issued precious-metal specie made by the US Mint. There had been widespread exchange of paper “bank notes”, which were transferable promissory notes of hard currency issued by individual banks, but the system was an ad hoc checker-board tied to individual institutions in both North and South. And because banks were far more uncertain propositions in the 19th Century than they are today, this bank notes carried a significant amount of risk as a store of long term value. With the start of hostilities, this risk compounded, and at the initial phases the entire financial system of the South (and even to some degree the North) was driven back on specie.
This was a gigantic problem for the Southern economy, because they had disproportionately little of the nation’s supply of specie. America’s banks were largely in the North, while Southerners held the vast majority of their wealth in land and (especially) slaves. So at the start of hostilities the government of the Confederacy faced not only the problems of financing a war and building a treasury from scratch, but also providing a sufficient money supply for basic commerce to continue. This lead rapidly to the South beginning to issue paper notes promising the repayment on one US dollar after the war. (Promising repayment in a foreign currency had precedent from the American Revolution, as similar paper notes issued then had promised postwar payment of one Spanish Dollar, which is partly how US currency came to take its name from outside the British tradition.)
In addition to this the CSA was heavily dependent on foreign sources of arms and munitions, particularly before the North’s blockade began to bite deeply later in the war. Foreign markets would absolutely not accept Confederate paper, so the Confederacy had to either barter (usually in cotton) or pay cash (or in this case gold) on the barrel. At the same time, the people of the Confederacy actually placed a “cotton embargo” on Europe in 1861 to try and coerce European intervention. It was a strange, but very clear and powerful, popular movement as it was conventional wisdom that starving Britain and France of cotton was guaranteed to pull them into the conflict. This diplomatic effort failed disastrously on several levels, and the CSA sold every ounce of their 1862 cotton crop they could sneak out of the South. But it did mean that pretty much all of the Confederacy’s 1861 import purchases had to be made from an already-inadequate stock of hard currency.
Confederate Secretary of the Treasury Christopher Memminger recognized that his first task was getting as much hard currency into the hands of the government as possible. His first (and probably clearest) success was Louisiana’s decision to join the Confederacy in March 1861. They brought with them the US New Orleans Mint, one of a handful of locations where the US government had minted coins and stored gold bullion. The already-minted US coins there were seized (and spent as was) and the stock of bullion was minted out using new dies as official Confederate specie.
(One important digression here is that exchange rates between currencies worked somewhat differently in the era of precious-metal specie. Part of this resolves around the confused popular notion, both then and now, of why currency is valuable. People thought a coin was valuable because if it’s precious metal content. So exchange rates could never drift very far from a narrow band of things being “worth” the weight of precious-metal you would get from melting them down. And because this brutally direct method of arbitrage existed, determining relative values of currencies was actually fairly simple and could be mostly evaluated by a good scale to weigh them. This is why there were few administrative difficulties or stigmas from using foreign currency in this time period, and why the US gold dollar remained easy to spend in the South throughout the war.)
After the boon Memminger received from the New Orleans Mint, nothing else came so easily. The South had some banks, and some wealthy individuals with hard currency, but direct appropriation was not administratively realistic and would have caused more systemic harm than good. So Memminger first moved towards issuing a fiat (i.e., unbacked paper) currency to provide a functional money supply, the issuance of debt, and systems of taxation to pay for the war. He began by trying to issue war bonds, but after an initial patriotic surge, wealthy Southerners became reluctant to invest their dwindling hard money in the cause. So then he focused on the traditional American form of taxation, import tariffs. (Antebellum Americans did not actually pay the Federal Government direct taxes in many ways. And tariffs had the advantage of being arms length from the average voter, so it fostered less resentment.) But this source of revenue was undercut by popular opposition to tariffs (opposition to the “Protective Tariff” was the only other specific Southern policy grievance leading to war after slavery), and the fact that the Union blockade was destroying the import market anyway. So then Memminger moved to the unprecedented policy of imposing direct income taxes. But in the face of no enforcement apparatus and state governments who were (unmetaphorically) violently opposed, this too was totally inadequate.
So Sec. Memminger adopted the only policy left that he could find: printing fiat money. That is, he went beyond providing enough paper currency to keep a functional money supply, and instead simply started printing paper CSA dollars to pay for all government expenses. Predictably, this lead to ruinous inflation and the collapse of the already-weak Confederate economy. But it also lead to a huge surge in demand by Southerners for US dollars.
While all this had been taking place in the South, the North had adopted similar steps; they had issued fiat paper currency and imposed a direct income tax. But the new US paper currency, the “green backs” that began the tradition of all US money being green, was not simply printed at whim. Instead it was issued on the basis of private bank reserves stored at the US Treasury. Because issuance was limited, the greenback retained much of its theoretical “gold equivalent” value throughout the war. This meant that, by late in war when the rump Confederacy was racked by hyperinflation, Southern cotton smugglers working with Northern “agents” would happily accept paper northern money over no money at all. There were few quotidian commercial transactions in the South by that point, but when they did occur they were often in US currency.
TLDR prewar US currency was all in precious metals, so it retained most of its value and usefulness after the start of hostilities, though it had a high chance of leaving the Confederacy to pay for needed imports. During the war, disastrous Confederate fiscal and monetary policy effectively destroyed the economy of the CSA. Northern monetary policy was far better handled, so even US paper currency continued to be valuable (and used) in the South for much of the conflict.