Say we have an involutive vector subbundle $\pi:E\subset TM\to M$, where $\pi:TM\to M$ is the tangent bundle to a paracompact smooth manifold $M$. Then $E_p=\pi^{-1}(\{p\})$ is the tangent space to each leaf $L$ of the foliation determined by $E$, for each $p\in M$.
Now if we have a linear connection $\nabla^L$ defined on each leaf, that is, $\nabla^L$ is a $\mathbb{R}$-bilinear map
$$ \nabla^L:\mathfrak{X}(L)\times\mathfrak{X}(L)\to\mathfrak{X}(L) $$ that is $C^\infty(L)$-linear in the first argument and satisfies Leibniz' rule (i.e., $\nabla^L$ is a linear connection restricted to the subbundle $E$).
Can we obtain a "full" linear connection $\nabla$ on $TM\to M$ from such family of linear connections that restricts to $\nabla^L$ at each leaf of the foliation?
This seems like a partition of unity argument, but it's not clear how to construct such $\nabla$. Furthermore, uniqueness seems unlikely in this case.