At the birth of federation, Starfleet presumably have a mix of ships designed by the four members, Earth, Vulcan, Andor, Tellar, and for some time operates all those ships, .
We don't know much about Tellarite ships. But vulcan ships are clearly much more advanced that anything earth has, which is only natural as vulcans have been warp capable for centuries : the monasteryof P'Jem was build in the 9th century BC (long before surak, so it wasn't a "logic" monastery) and was in another star system, probably requiering warp (unless they made decades-long travels at relativistic speeds?). Vulcans ships go to warp 7 (only one warp index slower than Kirk's ship), they have shields, tractor beams...
Andorian ships are not as advanced as vulcan ones, but still superior to NX-01.
But, strangely, Kirk's and Picard's enterprises clearly are the grandchildren of NX-01, not of Vulcan or Andorian designs. So the whole question is : what happened ?
My guess :
At the beginning, starfleet kept shipyards in all four systems, each producing ships of the characteristic design lineage of this planet.
But vulcans kept their own fleet (there are still non-starfleet vulcan ships in TNG era) so their fleetyards were not 100% available for starfleet, while Earth's growing shipyards capacity was. Also, with time, it was realized that the terran basic design (with the two warp nacelles) was very versalite, and, once upgraded with vulcan technology, proved very efficient.
So, gradually, the other design lineages were dropped, and the earth-sources design principles were adopted, and finally, ships following human basic layout (but designed with vulcan and other engineers) were built even in the shipyards in other systems.
To answer the initial question, for me, there are at least two intermediate ship classes between NX and constitution.
