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).

These papers may perhaps not quite have the focus you're asking for: rather than deduce information from the moment polytope alone, they are about enhancing it with extra data (the Duistermaat-Heckman measure, a genus and a "painting") so that the resulting invariant completely determines the $T$-space.

The third one 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