covering maps in fiber bundles

(This question may be related to a previous one on the universal cover of Grassmann, but the relation is not clear to me.)

I have a fiber bundle pi:E-->B where E and B are smooth connected manifolds.

Does there always exist a smooth connected submanifold M of E with the following property, denoting by pi_M the restriction of pi to M:
for every x in B there exists an open neighborhood U such that the inverse image of U under pi_M is a union of at least one and finitely many mutually disjoint open sets of M each of which is mapped diffeomorphically onto U by pi_M?


(The example I have in mind is the Grassmann manifold. View it as O(n+k)/(O(n) x O(k)) or as ST(n+k,k)/GL(k) where ST(n+k,k) is the set of all full-rank (n+k)xk matrices. At least for ST(n+k,k)/GL(k) I know that there is no "M" as specified above with the additional requirement that M is invariant by the left action of O(n+k) on ST(n+k,k). But if I don't require this invariance, I don't see why such an M would not exist. I just don't know how to construct one.)

Thanks!
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