When the US Supreme Court "overturns" a precedent, it does not usually mean that every case which previously cited the older holding instantly vanishes or is upset. Indeed, often the older case is not truly overturned, but "distinguished", meaning the circumstances of the new case are different enough that the old rule does not apply. But, at least in theory, a case exactly similar to the old one would still follow the old rule. Sometimes such "distinguishing" cases pile up until nothing is left of the old rule.
Even when one case overturns an earlier one more dramatically, it may not be instantly total. For example, West Coast Hotel Co. v. Parrish (1937) directly overturned Adkins v. Children's Hospital (1923) and by implication Lochner v. New York (1905) But the tide had started to turn against Lochner some years earlier in Nebbia v. New York (1934), and Lochner continued to be cited favorably in some cases until United States v. Carolene Products Co. (1938). In Lincoln Federal Labor Union v. Northwestern Iron & Metal Co (1949) Justice Black (appointed 1937) wrote that the Lochner line of cases had been "repudiated".
A more dramatic case of an overturned precedent was West Virginia State Board of Education v. Barnette, 319 U.S. 624 (1943) (the 2nd flag salute case) which directly ruled the exact opposite of Minersville School District v. Gobitis, 310 U.S. 586 (1940). The membership of the court was largely the same, and the time lapse quite short. The Court essentially changed its collective mind, largely adopting the view of the dissent in the earlier case.
Another example of an extreme reversal is Dred Scott v. Sandford (1857), which denied that any black could be a US citizen. This was overturned by the outcome of the US Civil War, and the 13th, 14th, and 15th Amendments.
In general when an earlier case is overruled, the change is not retroactive. Old cases are not undone. But new cases will be less likely to cite the overturned case (except possibly as a bad example), and it will no longer be considered good law to mandate a decision in a lower court. This is particularly true if several new cases, over a period of time, follow the new rule or principle.