“Chabadniks put on Rashi tefillin first, then, near the end of Shachris, put on Rabbeinu Tam tefillin. I know this practice as encouraged by Rebbe Menachem Mendel Schneersohn, but I do not know whether it started before he became Rebbe.”
Not near the end of Shachris – we do it at the very end after tehillim which we say after ach tzadikkim.
The minhag started after the Rebbe became rebbe in 1950. People would ask special permission to put on the rebbe sometimes gave, but only starting from married and onwards. then, in 1976, the Rebbe formally announced that everyone should start putting on rabenu tam starting from 2 months before BM by hanochos tefillin.