first of all I think your $S(F)$ can be modified into \begin{align*} S(F)(t,x)=(t-\log(|f(x)|), F(x)) \end{align*} since $f$ is non-vanishing, this is always smooth. Nevertheless, there is a more conceptual way to see the symplectization: the symplectization $S$ is a functor from contact manifolds into homogeneous symplectic manifolds. The latter is the category of pairs $(P,\omega)$ consisting of a $\mathbb{R}^\times$-principal bundle $P$ and a symplectic structure $\omega\in \Omega^2(P)$, such that \begin{align*} h_r^*\omega=r\omega \end{align*} for the principal action $h\colon \mathbb{R}^\times\times P\to P$. The morphisms are equivariant symplectomorphisms. This functor is even an equivalence of categories and does not work just for co-orientable contact structures. Everything what I said is (more or less) done in Remarks on Contact and Jacobi Geometry (Bruce, Grabowska, Grabowski 2015).
HD
