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.
If you count the 22nd century scenes of "TATV(2161)" there are ninety-three years between the decommissioning of the NX-01 and "The Cage." A full century plus change if you have to wait until Kirk's five year mission commences.