Let $\pi:X \rightarrow Y$ be a surmersion (surjective submersion) between closed manifolds.

1) Is there any obstruction to the existence of a "multi-valued" section $s$ of $\pi$ such that $\pi \circ s$ is a smooth covering of $Y$ ?

By a multi-valued section I was thinking about gluing local section of $\pi$, where by a local section I mean a map $\sigma : U \rightarrow X$ with $U$ and open subset of $Y$ and satisfying $\pi \circ \sigma = id_U$.  The multi-valued section should take the form of an immersion $s : Y' \rightarrow X$ with $p: Y' \rightarrow Y$ is a covering map and such that $\pi \circ s = p$.

2) Is it always possible to restrict us to **finite** covering $p : Y' \rightarrow Y$ ?

A trivial example is given when the fiber bundle is trivial and the covering is just the trivial covering.