The subgroup $\hat T$ cannot contain any integral points because each $t \in \hat T$ is the $t$-translate of the origin which is not an integral point.Therefore by Nagell-Lutz $\hat T$ is either trivial or has exponent $2$, and I think it's known that there can be at most one non-integral torsion point $t \neq 0$, so $\hat T$ must have order $2$.
As Joe Silverman notes in a comment, if the curve has rank zero then automatically $t \in \hat T$ so $\hat T$ is nontrivial. If moreover $t$ is the only nontrivial torsion point then $\hat T = T$. Such examples are now easy to find in the LMFDB by asking for rank 0, torsion Z/2Z, and zero integral points; and there are plentiful examples starting from the curves 14.a1, 15.a1, 15.a3, ...
But this may not be deemed interesting because there are no integral points to permute. If there are integral points then they must come in sets of 4. The first example of torsion Z/2Z and four integral points is the curve 274.b2: $y^2 + xy = x^3 - x^2 + 8x - 6$, with 2-torsion point $t : (x,y) = (3/4,-3/8)$ and pairs of integral points at $x=1$ and $x=35$ that are switched by translation by $t$. The next example(285.b1) works too.