I have only a passing familiarity with this subject, but I believe the notion of a uniform national currency had only begun to develop by the time the Civil War began, as most banking functions, including the issue of bank notes were still issued by state-chartered banks, not federally-chartered or actual federally-operated central banks. Given this, centrally-issued Federal currency (and its Confederate counterpart) would have been only a minor fraction of the money in circulation after the Civil War, so it's not as if all money circulating in the south was instantly worthless, because there were many different currencies in circulation as well as instruments that can perform exchange-of-value functions similar to currency such as letters of credit, IOUs, precious metals and so on. Not to mention good old barter and trade.

Before, during, and after the civil war, the union was already making some big structural moves to consolidate the currency from state banknotes into Federal banknotes. The most significant of these were the national banking acts of 1863 and 1864, which among other things introduced a ten percent tax on all state banknotes, exchanging them for federal notes at 90% of the state note's face value when they were used to pay for any federal services, taxes, or to borrow money. This tax did not apply to notes issued by the new "national banks" also introduced by the banking acts. National banks, while privately owned and operated, received their charter from the federal government instead of the state.

I cannot answer your specific question about what people did with their actual "Confederate" money, as in notes issued by the central bank of the Confederacy, but I expect someone much more familiar with the Reconstruction era can help with that. My guess is that an official arrangement was made to buy those confederate notes at some discount to face value, similar to state banknotes, as well as the spontaneous appearance of many "unofficial" ad-hoc secondary markets on which they could be liquidated at various rates.