I have been piecing through your question on secondary delegation of contract administration. I believe the answer will be that specific inclusions of CAGE and DoDAACs will be specific to local procedure or the plain language of the contract. First I reviewed the FAR, the DFARS, and the PGI. The DFARS and PGI provide no additional clarification beyond what is in  FAR 42.202(e). FAR 42.202(e) (1) has three requirements for secondary delegation; must be in writing, clearly state specific functions to be performed, and include pertinent contractual and other necessary documents.  Diving deeper, DCMA Manual  2501-01 (March 24, 2019) has an excellent explanation of other places of manufacture (paragraph 3.7). This paragraph explains a common secondary delegation of contract administration (your use of PoP suggests to me that you are familiar with this document.) Like the FAR, the manual is also silent on what specifically must be written out and what is required to be included in the contract. I follow the ACO's logic in the question you submitted. Contract administration has already been delegated to the contract administration office and no further delegation is needed beyond the inclusion of the CAGE code where performance or additional performance is being conducted. If the government (program office, contracting shop, primary CAO, and secondary CAO) and the contractor (at primary and secondary places of performance) understand the administrate functions to be performed, I would argue that no further clarification is required.