Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of this paper. 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 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. So presumably one could use this classification to determine if there are smooth actions.

Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of this paper. 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 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. So presumably one could use this classification to determine if there are smooth actions.

Addendum: Oops, it looks like there's some gaps in McGavran's arguments that were fixed by Hae Soo Oh at least in dimensions 5 and 6. Moreover, Oh's classification is in the smooth category. Moreover, in Remark (4.7) of this paper, Oh constructs examples of $T^n$ actions on simply-connected $n+2$-manifolds in all dimensions. Since he's working in the smooth category, this seems to show that there are smooth examples (although I don't see immediately in his argument where smoothness is proven).