Conversion of folates to polyglutamate derivatives. (6S)-5,6,7,8-tetrahydrofolyl-(gamma-L-Glu)(n) + ATP + L-glutamate = (6S)-5,6,7,8-tetrahydrofolyl-(gamma-L-Glu)(n+1) + ADP + H(+) + phosphate Cofactor biosynthesis; tetrahydrofolylpolyglutamate biosynthesis. Belongs to the folylpolyglutamate synthase family.