The difference $\frac12\dim(X)-\dim(T)$ is known as the ***complexity*** of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon  and Sue Tolman:

- [Centered complexity one Hamiltonian torus actions][1] (2001);

- [Complete invariants for Hamiltonian torus actions with two dimensional quotients][2] (2003);

- [Classification of Hamiltonian torus actions with two dimensional quotients][3] (2011).

The third paper also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.

  [1]: http://www.ams.org/mathscinet-getitem?mr=1852084
  [2]: http://www.ams.org/mathscinet-getitem?mr=2128388
  [3]: http://arxiv.org/abs/1109.6873