I was always under the impression that with the exception of cargo ships the NX-01 was the first Starfleet ship.
I was always convinced that the only other ship (again with the exception of cargo ships) was the NX 02 (at that point in time).
But upon re-watch of the season 2 finale we see several Starfleet ships (not of the NX class) come to the aid of enterprise during the Klingon attack.
So where do these ships fit in to Starfleet history? And they are Starfleet ships as they were wearing the uniform.
They're smaller, slower and older ships. IIRC, "First Flight" established they would have had a top speed of warp 1.9. The warp 5 Enterprise was the first ship that was technologically able to go on long-term, long distance exploration missions.
Weirdly, they didn't have hull numbers. I guess someome realized, "Oh shit! These are meant to be older ships but we've already started numbering at 01!"