The origin of haplogroup T seems to be in the Eurasian Plate, where all their basal branches have been found with the exception of a unique sample found close to the border. Basal branches of Haplogroup L and T have their highest diversity in Europe+Caucasus with echo in the Zagros and Taurus mountains.
Actually, Haplogroup T have not been found in Anatolian Farmers 8500 BP but only in the ancient Europe and their genomes are found to be closer to Upper Paleolithic HG than to Mesolithic HG or Anatolian Farmers.
If this T1a came from somewhere outside Europe I should think on a Upper Paleolithic survivors back migration but as they have not been found in Anatolia, I would guess that they came from somwhere between Black Sea and Dinaric Alps. Perhaps earlier than first LBK farmers, I will cite to Meyer 2015:
"Building on both the evidence previously available for the LBK and the evidence presented here, we suggest that the repeated occurrence of almost indiscriminate massacres, the possible abduction of selected members, and the patterns of torture, mutilation, and careless disposal all fit into the concept of prehistoric warfare as currently understood within anthropology. Particular LBK groups were singled out for as yet unknown reasons, attacked with brute force, and annihilated by others, probably close neighbors and very likely other LBK groups of the wider region. As has been shown, even within the overall quite homogenous-appearing LBK, recognizable boundaries did exist in many places. These borders most probably were a result of the spread of different groups without close social or biological kinship ties to one another who came in to close contact as a consequence of the LBK colonization pattern. In fact, because the LBK was the first complete Neolithic culture in Central Europe, today all farmers of this time and region are classified as members of the LBK by default, regardless of how these people defined themselves and how they differentiated themselves from their contemporaries."