Any new televised TREK in the forseeable future will involve a different ship and crew...and probably be set closer to Kirk's time because of the new movie.
Between Archer's and Kirk's time? Perhaps. How many years of Trek history is that?
But it would preclude much intraTrek guest starring that picking up from the end would allow.
There's a little more than a century between the end of ENT and the beginning of TOS (just under a century if you include "The Cage"). Plenty of room for good, quality Trek stories.