Depends on what you're describing as a "Confederate" Half Dollar. in 1861, the New Orleans Mint struck halves for 3 separate governmental entities: The US Government, the Louisiana State Government, and the Confederate Government. All were the identical, "usual" Seated Half design. Under Federal control, 330,000 were struck. Under Louisiana control, just over 1.2 million were struck, and under Confederate control almost a million more. So, if you own an "original look" 1861-O Seated Half, the odds are it wasn't struck under Federal authority.
The Confederacy *did* produce Halves in 1861 with their own bespoke reverse die. 4 of them, to be exact. Yours isn't one of the 4.

In 1879, a man by the name of Benjamin F. Taylor, formerly the Chief Coiner of the Confederate Mint, sold the original die used to produce these 4 Halves. It ended up in the hands of J.W. Scott, a New York coin dealer. Scott proceeded to strike 500 one-sided white metal impressions of this die, with advertising for his business on the "obverse." He also obtained 500 real 1861 Halves, planed the reverse devices off, and restruck the reverse using the Confederate die. These are found in varying grades - Scott's "donor" coins were both circulated and Mint State - and graded by the reverse only as the obverses were flattened by the restrike, making them ungradable. They are lighter than original Halves, having lost metal when planed. A very few (4 or 5) trial strikes were not planed, and show details of the original Federal reverse under the Confederate devices.
Scott Restrikes are quite valuable in the market today, going for thousands of dollars.
Various and sundry "remakes" have been made in the years since. They come in varying quality - some even had the wrong number of buds on the left of the reverse, 17 rather than the correct 19 - but are generally easy to identify.