Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of [this paper][1]. However, I haven't checked that the action is smoothable. The authors are only considering locally smooth actions in the paper (see [p. 170 of Bredon][2] for the definition of locally smooth); however, the construction yielding Theorem 4.7 is quite explicit, and seems to be at least piecewise-smooth. All such actions were subsequently [classified by McGavran][3]. So presumably one could use this classification to determine if there are smooth actions. 


  [1]: https://msp.org/pjm/1974/53-2/p12.xhtml
  [2]: http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf
  [3]: http://www.ams.org/journals/tran/1979-251-00/S0002-9947-1979-0531977-5/home.html