Catalyzes conversion of folates to polyglutamate derivatives allowing concentration of folate compounds in the cell and the intracellular retention of these cofactors, which are important substrates for most of the folate-dependent enzymes that are involved in one-carbon transfer reactions involved in purine, pyrimidine and amino acid synthesis. (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 A monovalent cation. Cofactor biosynthesis; tetrahydrofolylpolyglutamate biosynthesis. Belongs to the folylpolyglutamate synthase family.