I've always assumed that Kirk inherited the Enteprise which was largely unchanged from how it was under Pike's command (and which had a crew of about 200).
The encounter with the barrier just trashed the ship, and as a result, while the ship was able to LIMP back to Federation space using scavenged components from a mining facility, the damage was very severe and the ship had to undergo a full refitting.
I've always said that the "WNMHGB" ship was still the "all military" version of the ship, a heavy cruiser mainly dedicated to security operations, not long-range exploration. There were likely quite a few of these, not merely twelve.
But twelve were refit and relaunched as "explorers." The advent of replicator technology (food slots, quartermaster department requirements, spare parts manufacturing, etc) freed up a large amount of what was previously cargo space aboard this ship, and they basically put a full science vessel's complement of personnel and facilities into the freed up space. The Enterprise, and twelve other heavy cruisers, became the first "exploratory cruisers" and were simultaneously launched on a widely-publicized "five year exploratory mission" with great fanfare. When the Enterprise was the only ship to return, fully intact, from that mission, Starfleet elected to make the Enterprise insignia that of the entire Starfleet. (That, and I'm sure they were getting sick of coming up with custom insignia for every single ship they launched!)
So, there was likely at least a year between WNMHGB and The Corbomite Manuever. While the outside of the ship was mostly unchanged, the internal configuration was altered almost as dramatically as it was during the eighteen months prior to ST-TMP.