Existance of tubular neighborhood

Let $S\subset \mathbb{R}^n$ be a compact oriented hypersurface of class $C^k$, $k\geq 2$. There is a neighborhood $V$ of $S$ in $\mathbb{R}^n$ and a number $\varepsilon >0$ such that the map $F(x,t) = x+t\nu(x)$ is a $C^{k-1}$ diffeomorphism of $S\times(-\varepsilon, \varepsilon)$ onto $V$.

This statement and a "proof" can be found in Introduction to PDEs by Folland. I am looking for a more rigorous proof of this statement. Here is what I tried so far.

$F$ is $C^{k-1}$ since $\nu$ is. For $x\in S$ we want to look at the Jacobian of $F$ (w.r.t. local coordinates) at $(x,0)$ so we define a local parameterisation. There exist open sets $U\subset \mathbb{R}^{n-1}$, $x\in W\subset \mathbb{R}^n$ and a function $\psi\in C^k(U;\mathbb{R})$ so that $\psi: U\rightarrow S\cap W$ is bijective and has a continuous inverse. Local coordinates of $x$ are thus given by $u_0 = \psi^{-1}(x) \in \mathbb{R}^{n-1}$. Next we set $$G(u,t) = F(\psi(u),t)$$ for $u\in U$ and $t\in \mathbb{R}$. Clearly $G^{\prime}(u_0,0)$ is nonsingular since the first $n-1$ columns of $G'(u_0,0)$ consist of vectors that span the tangent space of $S$ at $x$ and the last column is just $\nu(\psi(u_0)) = \nu(x)$ which is orthogonal to said tangent space.

We now want to apply the inverse mapping theorem. It states that we can find open neighborhoods $(U_x\times I_x) \subset \mathbb{R}^{n-1}\times \mathbb{R}$ containing $(u_0,0)$ and $W_x$ containing $G(u_0,0) = x$ such that $G: U_x \times I_x \rightarrow W_x$ is a $C^{k-1}$ diffeomorphism. Hence $$F_x^{-1}: W_x \rightarrow \psi(U_x)\times I_x$$ is a $C^{k-1}$ inverse of $F$ (locally near $x$). I don't really know where to go from here as it is just a bit different of what's done in the "proof". For reference:

Hence by the inverse mapping theorem $F$ can be inverted on a neighborhood $W_x$ of each $(x_0,0)$ to yield a $C^{k-1}$ diffeomorphism $$ F_x^{-1}: W_x \rightarrow (S\cap W_x) \times(-\varepsilon_x,\varepsilon_x) $$ for some $\varepsilon_x>0$.

What I don't understand is why $F_x^{-1}$ maps to $(S\cap W_x)\times(-\varepsilon,\varepsilon)$. Specifically, where does this interval come from, why is is symmetric around $0$? Why do I only get these sets $I_x$ and $U_x$?

I kind of understand the next steps of the proof. By compactness you can choose $(x_n)_n\subset S$ such that $W_{x_n}$ cover $S$ and you somehow patch $F_{x_n}^{-1}$ together to receive a $C^{k-1}$ inverse of $F$.