In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$.

Basically, the reason is this: If $V$ is a $C^r$ vector field on $\mathbb{R}^2$ and $V(0,0)\not=0$, then there exist local $C^r$ coordinates $(x^1,x^2)$ centered on $(0,0)$ such that $V = \partial/\partial x^1$. Any other set $(y^1,y^2)$ of such $C^r$ 'flowbox' coordinates is locally of the form $$ (y^1,y^2) = \bigl(x^1 + f(x^2), \ g(x^2)\bigr) $$ for some $C^r$ functions $f$ and $g$ of one variable, that satisfy $f(0)=g(0)=0$. Generally, if $(x^1, x^2)$ is only $C^r$, there will not exist such functions $f$ and $g$ that will make $y^1$ and $y^2$ be $C^{r+1}$.

To construct a specific example, consider the vector field $$ V := a(x,y)\,\frac{\partial}{\partial x} + 0\,\frac{\partial}{\partial y} $$ where $a(0,0)\not=0$ and $a$ is only $C^r$, and let $u$ be the solution of the first-order PDE with initial condition $$ u_x(x,y) = \frac{1}{a(x,y)} $$ and $u(0,y) = 0$. Then $\bigl(u(x,y),y\bigr)$ will be a $C^r$ flowbox chart for $V$. Note that $u$ will only be $C^r$ in general. (Though it will have a $C^r$ first derivative with respect to $x$, its $y$-derivative will usually be only $C^{r-1}$.) In particular, there will not exist $C^r$ functions $f$ and $g$ of one variable such that $u(x,y)+f(y)$ is $C^{r+1}$.

For a more specific example, let $a(x,y) = 1 + x h(y)$ where $h$ is $C^r$ but not $C^{r+1}$.

In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$.

Basically, the reason is this: If $V$ is a $C^r$ vector field on $\mathbb{R}^2$ and $V(0,0)\not=0$, then there exist local $C^r$ coordinates $(x^1,x^2)$ centered on $(0,0)$ such that $V = \partial/\partial x^1$. Any other set $(y^1,y^2)$ of such $C^r$ 'flowbox' coordinates is locally of the form $$ (y^1,y^2) = \bigl(x^1 + f(x^2), \ g(x^2)\bigr) $$ for some $C^r$ functions $f$ and $g$ of one variable, that satisfy $f(0)=g(0)=0$. Generally, if $(x^1, x^2)$ is only $C^r$, there will not exist such functions $f$ and $g$ that will make $y^1$ and $y^2$ be $C^{r+1}$.

For a specific example, let $h:\mathbb{R}\to\mathbb{R}$ be a $C^r$ function that is not $C^{r+1}$ and satisfies $h(0)\not=0$ and consider the vector field $V$ defined on a neighborhood of $(0,0)$ in the $uv$-plane by $$ V := \frac{1}{h(v)}\,\frac{\partial}{\partial u} + 0\,\frac{\partial}{\partial v}. $$
The $C^r$ local flowbox coordinates for $V$ on a neighborhood of $(0,0)$ are of the form $$ (x^1,x^2) = \bigl(\ u\,h(v)+f(v),\ g(v)\ \bigr) $$ where $f$ and $g$ are $C^r$ functions. Clearly, it is not possible to choose $f$ and $g$ so that $x^1$ will be of class $C^{r+1}$.

In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$.

Basically, the reason is this: If $V$ is a $C^r$ vector field on $\mathbb{R}^2$ and $V(0,0)\not=0$, then there exist local $C^r$ coordinates $(x^1,x^2)$ centered on $(0,0)$ such that $V = \partial/\partial x^1$. Any other set $(y^1,y^2)$ of such $C^r$ 'flowbox' coordinates is locally of the form $$ (y^1,y^2) = \bigl(x^1 + f(x^2), \ g(x^2)\bigr) $$ for some $C^r$ functions $f$ and $g$ of one variable, that satisfy $f(0)=g(0)=0$. Generally, if $(x^1, x^2)$ is only $C^r$, there will not exist such functions $f$ and $g$ that will make $y^1$ and $y^2$ be $C^{r+1}$.

To construct a specific example, consider the vector field $$ V := a(x,y)\,\frac{\partial}{\partial x} + 0\,\frac{\partial}{\partial y} $$ where $a(0,0)\not=0$ and $a$ is only $C^r$, and let $u$ be the solution of the first-order PDE with initial condition $$ u_x(x,y) = \frac{1}{a(x,y)} $$ and $u(0,y) = 0$. Then $\bigl(u(x,y),y\bigr)$ will be a $C^r$ flowbox chart for $V$. Note that $u$ will only be $C^r$ in general. (Though it will have a $C^r$ first derivative with respect to $x$, its $y$-derivative will usually be only $C^{r-1}$.) In particular, there will not exist $C^r$ functions $f$ and $g$ of one variable such that $u(x,y)+f(y)$ is $C^{r+1}$. It is easy to write down specific examples.

In dimension $1$, it's true that a flowboxing change of coordinates for a $C^r$ vector field is $C^{r+1}$, but this is no longer true in dimensions greater than $1$.

Basically, the reason is this: If $V$ is a $C^r$ vector field on $\mathbb{R}^2$ and $V(0,0)\not=0$, then there exist local $C^r$ coordinates $(x^1,x^2)$ centered on $(0,0)$ such that $V = \partial/\partial x^1$. Any other set $(y^1,y^2)$ of such $C^r$ 'flowbox' coordinates is locally of the form $$ (y^1,y^2) = \bigl(x^1 + f(x^2), \ g(x^2)\bigr) $$ for some $C^r$ functions $f$ and $g$ of one variable, that satisfy $f(0)=g(0)=0$. Generally, if $(x^1, x^2)$ is only $C^r$, there will not exist such functions $f$ and $g$ that will make $y^1$ and $y^2$ be $C^{r+1}$.

To construct a specific example, consider the vector field $$ V := a(x,y)\,\frac{\partial}{\partial x} + 0\,\frac{\partial}{\partial y} $$ where $a(0,0)\not=0$ and $a$ is only $C^r$, and let $u$ be the solution of the first-order PDE with initial condition $$ u_x(x,y) = \frac{1}{a(x,y)} $$ and $u(0,y) = 0$. Then $\bigl(u(x,y),y\bigr)$ will be a $C^r$ flowbox chart for $V$. Note that $u$ will only be $C^r$ in general. (Though it will have a $C^r$ first derivative with respect to $x$, its $y$-derivative will usually be only $C^{r-1}$.) In particular, there will not exist $C^r$ functions $f$ and $g$ of one variable such that $u(x,y)+f(y)$ is $C^{r+1}$.

For a more specific example, let $a(x,y) = 1 + x h(y)$ where $h$ is $C^r$ but not $C^{r+1}$.