While people typically see the Civil War as a war about slavery, it can also be seen as a war about the South’s right or ability to govern itself as it saw fit. When we look at the Civil War in this second way, we can see how Reconstruction is a continuation of the war.
After the Civil War, the South started to try to assert its right to self-rule again. It did not try to reinstate slavery, but it did institute the “black codes” to try to control the newly freed African Americans. The Radical Republicans felt that the South was trying to ignore the outcome of the war and they eventually imposed military rule on the South. This time of military rule was a time when the North was trying to force the South to have a certain kind of government, one which treated African Americans equally. This can be seen as an extension of the Civil War because it involves the North trying to prevent the South from ruling as it liked.
Throughout this period, the Southerners resisted Northern rule in a variety of ways. They did not stop resisting until they “redeemed” the South. We can see the end of Reconstruction as the end of the Civil War because it ended the North’s attempt to force the South to have certain kinds of laws.