Short question with long title:

Suppose $T$ is the tangent functor and $T^2:=T\circ T$ is the second order tangent functor.

Are there natural transformations $T\otimes T \Rightarrow T^2$ ?

I consider the 'usual' domain of $T$ and $T^2$ as the category $\mathbf{M}$ of finite dimensional Hausdorff manifolds together with smooth morphisms, but in case there is a difference feel free to restrict to the category $\mathbf{M}_n$ of $n$-dimensional Hausdorff manifolds together with local diffeomorphisms.