The inverse function theorem is applied for function $f: \Bbb R^n \to \Bbb R^n$.
Question : What is the natural generalization of inverse mapping theorem to the functions $f: \Bbb R^n \to \Bbb R^m$ ?
And how one can prove this using traditional inverse mapping theorem?
Note that in latter case the Jacobian of $f$ is a $m \times n$ matrix, so we can not think about inevitability of Jacobian, but instead still we may impose the condition of being Jacobian full column rank.
The closest I can think of is the implicit function theorem. In contrast to the inverse function theorem, which you can apply at points $p$ with $\det (Df)_p\neq 0$, you can apply it on points where the differential is surjective (so especially only when $n\geq m$). Quickly said it states that if the differential of a continously differentiable function is surjective at a point, then the preimages of values near the attained one can locally be represented as graphs.
The exact formulation goes as follows: Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be a $C^1$ function and $p$ such that the differential $(Df)_p$ is surjective. (From now on wlog $p=0$ and $f(p)=f(0)=0$) Let $K:=ker (Df)_p=((Df)_p)^{-1}(0)$ be the kernel of the differential at that point and let $S$ be a complementary subspace of $\mathbb{R}^n$, i.e, $\mathbb{R}^n=K\oplus S$. Let $\pi:\mathbb{R}^n\to K, v=k+s\mapsto k$ be the projection on the kernel. Then there exist neighborhoods $U$ of $p(=0)$ in $K$, $V$ of $f(p)(=0)$ in $\mathbb{R}^m$ and a $C^1$ function $g:U\times V\to \mathbb{R}^n, (k,w)\mapsto g(k,w)$ with \begin{equation} f(g(k,w))=w \, \text{and} \,\pi(g(k,w))=k. \end{equation}
The interpretation is (as I had stated above) as follows: Fixing some $w_0$ not to far away from $f(p)$ one can represent the preimage of $w_0$ near $p$ as the a graph (over the kernel of the differential).
The more commonly used special case is the following: Let $f:\mathbb{R}^n\to \mathbb{R}$ be a $C^1$ function and $p$ such that $\frac{\partial f}{\partial x_{n}}(p)\neq 0$. Then there exists a neighborhood $U$ of $(p_1,...,p_{n-1})\in \mathbb{R}^{n-1}\times \{0\}\subset \mathbb{R}^n$, an $\epsilon >0$ and a $C^1$ function $g:U\times (f(p)-\epsilon,f(p)+\epsilon)\to \mathbb{R}^n$ with \begin{equation} f(x_1,...,x_{n-1},g(x_1,...,x_{n-1},z))=z. \end{equation} (To see that this is a special case draw a picture and set $S=\mathbb{R}e_n$.) Most of the time the theorem is only used for $z=f(p)$.
I hope this is a kind of generalisation you were looking for.
Remark: The inverse and the implicit function theorem are even equivalent even though the implicit function theorem seems more general.