Of course! Of course I'm sure you're familiar with the Starfleet pennant on the side of the TOS Enterprise's hull (and nacelles), which is the TOS Starfleet boomerang. It was also seen hanging on the walls of Starfleet command locations, like in in court room in "Court Martial" - [link]

It was also seen behind Starfleet admirals and commodores when they would speak to Kirk over subspace, like in this frame from "The Trouble with Tribbles" - [link]

In the TOS era, each starship had its own distinctive badge that members of its crew would wear. That way, you could tell what starship someone belonged to just by looking at their badge. Only USS Enterprise crewpeople wore the "arrowhead;" people from other ships wore other ship badges. For example...

USS Exeter badge (from "The Omega Glory"): [link]

USS Constellation badge (from "The Doomsday Machine"): [link]

USS Antares badge (from "Charlie X"): [link]

Even the seriously flawed (and in my opinion, non-canon) TV show "Enterprise" got this right, when they designed a specific badge for the 23rd century starship USS Defiant: [link]

Starfleet personnel who were NOT assigned to a starship, and were instead part of a starbase or other installation, would wear the "starburst" badge (even if they were in temporary command of a starship), seen here:

From "Court Martial" - [link]

From "The Ultimate Computer" - [link]

From "The Deadly Years" - [link]

As you can see, the boomerang was the proper Starfleet logo of the TOS era - only 430 people out of the thousands in Starfleet wore the arrowhead. I hope that helped!