It would help if we had a TOE for TF38. This is further complicated by TF38 actually being three separate units. One of which (TF38.2 maybe--fickle memory) having an amphibious assault force temporarily attached.
The primary expendable for a combat TF is fuel, so the primary support ship will be an AO--an oiler equipped for underway replenishment ("UnRep" is the USN term-of-art).
Any other needed ship would be sent separately to rendezvous with the TF at some other location.
The reason for that is in relative speed. A TF, particularly a carrier-based TF needs to operate at speed beyond "best composite fuel cruise speed"--this is especially true for Flight Operations, where 30kt speed is needful.
For WWII Amphib groups, those ere laden with support ships. Ships like LST, and LSI were particularly slow, with best-fuel cruise speeds of 9-10kt, which was slower than the 14-16kt speeds of APD, AK, AP, and the like. Or the 13-15kt Liberties (which was also why "Victory" ships were much preferred, as they cruised in the 14-18kt range).