My dad served on an oiler (USS Wabash) 44-46. After Japan surrendered he was assigned to a task force to clear mines off the coast of China. Was an oiler big enough to hold enough fuel so that it just stuck with the other ships and filled them as needed? Or was it constantly traveling back to port to refill it's tanks and rejoin the task force?
I found the definitive answer here:
https://www.history.navy.mil/research/library/online-reading-room/title-list-alphabetically/b/beans-bullets-black-oil.html
The TL;DR answer:
It was war, they did both, whatever was needed , it was WAR!
After reading the above , excellent, article I can add:
In my dad's case, his oiler Wabash was assigned long term to a task force (a group of lite mine sweepers) and his oiler just drifted along supplying fuel. The logistics developed and deployed during WWII were pretty amazing.