From a quick search on the web, using “Task Force 58, Spruance”:

“The Fast Carrier Task Force” came into being in late 1943, after the arrival in the Central Pacific of the first ships of the Essex and Independence classes. This force was the Pacific War’s equivalent of the great gunship battlefleets of earlier conflicts. By the time of the Battle for Leyte Gulf it had already proved itself to be one of the most potent instruments in the history of naval warfare; obliterating Japanese air power, and sweeping enemy warships and merchant shipping from the seas, wherever it had ventured.

It was divided into carrier task groups, each group containing typically between three and five carriers, and with each group having its own strong escort, a large number of cruisers and destroyers, and often two or more of the new fast battleships.

From early 1944 the Fast Carrier Force was known as ‘Task Force 58′ when serving under Admiral Spruance’s Fifth Fleet, and as ‘Task Force 38′ as part of Admiral Halsey’s Third Fleet. (Third and Fifth Fleets in general consisted of the same vessels, only their command teams differed).
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From a website devoted to the history of the USS Essex (lead ship of the Essex class).

As I remember, Spruance was in command at Midway because Halsey came down with some kind of itching hives.
Jay, with all due respect to your veteran friend, I think he’s gotten a few things messed up in his memory. Not saying he is forgetful or not sharp, just gotten a few things mixed up. I’m sure he is a tremendous guy. I once knew a whole bunch of WWII vets of all kinds (one retired as a two-star general in the Air Force, and flew P-38′s in North Africa; he’s now long dead), and they sometimes got things out of sequence in re-telling their stories. That doesn’t lessen my regard for any of them at all. But now they are almost all gone, including my Dad, who was a veteran of the Pacific War