There are numerous routes up Mt Moosilauke & nothing has emerged as a particular "FKT route".

Ben Nephew posted a double traverse option (shown on map):

http://www.hikenewengland.com/WelchDickey030719.php

This is a 4.5 mile loop over the summits of Welch & Dickey Mountains near Waterville Valley.

Stanislav Trufanov posted the route:

A Wild River Loop is a rim run of the upper Wild River Basin. Depending on one's goal, several versions of this loop may be done. I will attempt to establish some definitions but any feedback(& GPS data) is appreciated!

NOTE:  Sections of this trail are no longer accessible due to private land.  No new FKTs can be set, unless something changes.

The Narragansett Trail runs about 22 miles from Ashville Pond in Hopkinton, RI, to Lantern Hill in Ledyard, CT.  The 16- mile Connecticut section seems to be better documented.  A complete map is here:  http://auntiebeak.com/wp-content/uploads/2016/09/narr-trail-map.pdf