Originally, all of Main Street in Newington and South Street in West Hartford was designated Route 176; Hartford Avenue in Newington and Newington Avenue in Hartford were part of the original definition of Route 175.
In 1940, Route 175 was redefined to follow Cedar Street into Wethersfield and Hartford Avenue switched from Route 175 to Route 176. The leftover portion of Route 176, north of the Hartford Avenue turnoff, became Route 176A.
In 1962, the West Hartford portion of Route 176A was turned over to the city. In 1969, the Newington section was turned over. I don't know if Route 176A was partially signed during that period, or if signs were removed.