Let E be an elliptic curve over the rationals. Let L be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension L(E_tor) of the rationals which is generated by the set of x- and y-coordinates of the torsion points in E with respect to a Weierstrass model of E with rational coefficients. In this paper we will prove a lower bound for the absolute logarithmic Weil height of non-zero elements in L(E_tor) that are not a root of unity.