A person can be deprived of rights of citizenship without ceasing to be a citizen. For example, every person convicted of a felony loses certain rights of citizenship, such as the right to vote, the right to possess certain weapons, etc... at least temporarily. Some of those rights are restored automatically when the sentence is completed, others require application for restoration. Those sanctions are done by statute and are subject to challenge through the courts. If Congress wants to pass a bill that says all felons can vote, or possess a firearm, they could do it. However, the felon doesn't cease to be citizen of the United States.
In the case of Reconstruction, when they talk about "readmitting the state," it is shorthand for readmission of the state's representatives to Congress, not of the state to the United States at large. Each house of Congress has control over who is seated. Article I, Section 5 of the United States Constitution states in part: "Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members, in such Manner, and under such Penalties as each House may provide.
Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behavior, and, with the Concurrence of two-thirds, expel a Member."
Lincoln's 10% Electorate Reconstruction Plan in Louisiana in 1863 was hamstrung when Congress refused to seat their newly elected senators and representatives. A modern discussion of Article I, Section 5 came up recently in the context of the Alabama senate race with Roy Moore.