I have to prove the following fact. Let $\pi :E\to M$ be a vector bundle of rank $k$, over a $n$-manifold $M$. If $S\subseteq M$ is closed and $s\colon S\to E$ is a partial section of $\pi$ (i.e. $\pi(s(x))=x$ for all $x\in S$), then $s$ can be extended to a global section $\widetilde{s}: M\to S$.
My idea is to use the known fact (that is a consequence of the existence theorem of partitions of unity) that every smooth map $f: S\to\mathbb{R}^h$ can be extended to a global one $\widetilde{f}:M\to\mathbb{R}^h$.
Proof (idea). Let $\pi_2\colon M\times\mathbb{R}^k\to\mathbb{R}^k$, $\pi_2(p,v):=v$ the natural projection over the second factor. Let $\{\varphi_\alpha\colon U_\alpha\to V_\alpha\}_{\alpha\in A}$ be a smooth atlas of $M$ trivialising the bundle $\pi$, i.e. there is a local trivialisation $\chi_\alpha:\pi^{-1}(U_\alpha)\to U_\alpha\times\mathbb{R}^k$ of $\pi$. Let's consider the smooth map $g_\alpha\colon s^{-1}(\pi^{-1}(U_\alpha))\to\mathbb{R}^k$ given by $g_\alpha=\pi_2\circ\chi_\alpha\circ s_{|U_\alpha\cap S}$. Notice that $s^{-1}(\pi^{-1}(U_\alpha))=S\cap U_\alpha$ is open in $S$ (possibly empty). Furthermore, $S\cap U_\alpha$ is closed in $U_\alpha$ since $S$ is closed in $M$.Then for all $\alpha\in A$ the map $g_\alpha\colon S\cap U_\alpha\to\mathbb{R}^k$ can be extended to a global one $\widetilde{g_\alpha}\colon U_\alpha\to\mathbb{R}^k$. Now, my idea is to consider a partition of unity $\{\rho_\alpha\}_{\alpha\in A'}\cup\{\rho\}$ of $M$ subordinate to the open cover $\{U_\alpha\}_{\alpha\in A'}\cup\{M\setminus S\}$ of $M$, where $A':=\{\alpha\in A\mid S\cap U_\alpha\ne\emptyset\}$. Let's define $\widetilde{g}\colon M\to \mathbb{R}^k$, $\widetilde{g}:=\sum_{\alpha\in A'}\rho_\alpha\widetilde{g_\alpha}$. Now I don't know how to define $\widetilde{s}:M\to E$ in a suitable way. Can you help me please?
Take a local trivialization over an open set $U\subset M$ such that $U\cap S\not=\emptyset$, let $\chi:\pi^{-1}(U)\xrightarrow{\sim}U\times\mathbb{R}^k$ be the trivialization map.
The restriction of $s$ induces a section $s_{U,S}:U\cap S\to\pi^{-1}(U)$, using $\chi$ we identify $s_{U,S}$ with a function \begin{equation} \begin{split} \psi_{U,S}:=\chi\circ s_{U,S}:U\cap S&\to U\times\mathbb{R}^k\\ x&\mapsto(x,f(x)) \end{split} \end{equation} As you mentioned, this can always be extended to a map $\psi_U$ on the whole open set $U$, and $s_U:=\chi^{-1}\circ\psi_U$ is a section of $E$ on $U$ exteding $s_{U,S}$.
This is more or less the local construction, now one should work out how to make it global in order to extend $s$ to the whole of $M$. The idea I guess is to take a partition of unity $f_\alpha$ subordinated to an open cover $\{U_\alpha\}$ of $M$, and you can also assume that there are local trivializations of $E$ over these $U_\alpha$. Now you can take the new section to be $s_M=\sum_\alpha f_\alpha s_\alpha$, where $s_\alpha:=s_{U_\alpha}$ as defined earlier if $U_\alpha\cap S\not=\emptyset$, and $s_\alpha=0$-section otherwise.
There are probably a few things to check and tweak to make this work, let me know if this makes sense to you.