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 https://arxiv.org/pdf/1507.05405.pdf. 

HD