In this paper we present an alternative cosmological extension of the three-dimensional extended Newtonian Chern–Simons gravity by switching on the torsion. The theory is obtained as a non-relativistic limit of an enhancement and U(1)-enlargement of the so-called teleparallel algebra and can be seen as the teleparallel analogue of the Newtonian gravity theory. The infinite-dimensional extension of our result is also explored through the Lie algebra expansion method. An infinite-dimensional torsional Galilean gravity model is presented which in the vanishing cosmological constant limit reproduces the infinite-dimensional extension of the Galilean gravity theory.