Baird and Haartz do not list one for S. Kinsman although they do say that their list is "tentative." These various structures, some private, some public, came and went over time. In the local vicinity, there were at one time fire detection towers or other structures on Garfield, Cannon, Hale, Cooley Hill (Easton, base remnant remains), Mts. Agassiz and Rosebrook (Bethlehem), Moosilauke (summit buildings), and a "Grandview" in Woodstock.
But this might have something to do with it. An 1880's guidebook, which does not differentiate N. from S. Kinsman, remarks that it was tough getting to the summit of Kinsman. Easy enough climbing to the top of the slide but then a difficult route through "bad scrubs" after that. Once on top, though, there were signs of civilization, including the fact that the summit was "the repository of a U. S. Coast Survey signal."
I was curious about why a coast survey would be using Kinsman, so here's a little background on the US Coast Survey:
"The Office of Coast Survey is the oldest U.S. scientific organization, dating from 1807 when the United States Congress directed that a "survey of the coast" be carried out. In the ensuing years additional responsibilities were assigned to the young agency to meet scientific and engineering needs of a growing national population and economy and by1836, it was called the U.S. Coast Survey. In1871, a geodetic connection between theAtlantic and Pacific coasts was officially authorized and in 1878, the name was changed to the U.S.Coast and Geodetic Survey (C&GS). In 1926, the production of aeronautical charts was added to meet the requirements of the new air transportation age. In 1970, the U.S. Coast and Geodetic Survey became the National Ocean Survey under the newly established National Oceanic and Atmospheric Administration (NOAA) and in 1982, it was renamed the National Ocean Service (NOS). In December of that year, the Office of Charting and Geodetic Services was created within the new National Ocean Service to better focus on charting and geodetic activities, re-establishing the former C&GS acronym. Then in 1991, the office was renamed to its former 1878 name of Coast and Geodetic Survey."
If not a remnant of that coast survey signal, what you see on S. Kinsman
might be left over from a similar survey.