They were not "readmitted" to the the Union under the terms of Article III, Sec. 3 of the US Constitution because secession was never recognized so their original admission remained valid.
You are confusing admission to the Union with the US Congress agreeing to seat members from those states after the reconstructed state governments had complied with terms dictated by congress.
Legally, they were never out of the Union, but they were out of the Congress.
Legally, They left the Union
The Constitution is a VOLANTARY compact formed by the States.....
Whether them leaving unilateral is a violation of that agreement was vacated by Lincoln choosing force...