DY:thanks
yekke2:Your grandfather probably saw it in the Taamei Minhagim .The Magen Avraham actually gives a different reason he says that the Maharil had a special tallis and he used to fold right away in order to do a mitzvah right away and he said the people of tzippori(Berchas 53) would do the same.
I will look up the Rivevos Ephraim who talks about folding a tallis on shabbas.