= Solution
For a <coordinate chart> $x:U\to x(U)$ and either <orientation> sheet over $U$, take the chart $x\circ\pi$ on that sheet. On a nonempty overlap, its transition to $y\circ\pi$ is exactly $y\circ x^{-1}$, restricted to the open agreement set from part (a). It is smooth with smooth inverse. These charts therefore define a smooth structure for which $\pi$ is locally a <diffeomorphism>.
The topology is <Hausdorff>: points with distinct base points are separated by inverse images of disjoint base neighborhoods; two points over the same base point are separated by the two disjoint sheets over one coordinate neighborhood. A countable coordinate cover of $M$ gives a countable collection of two-sheeted charts, whose Euclidean chart bases show <second countability>. Thus the cover is a genuine <smooth manifold>, not just a collection of compatible charts. The construction using all base charts makes the smooth structure independent of the chosen atlas. It also makes $\pi:\widehat U_\omega\to U$ a <diffeomorphism> for every oriented open $U$, since it is a bijective local <diffeomorphism>.
At $\widehat p=(p,o_p)$, the derivative $d\pi_{\widehat p}:T_{\widehat p}\widehat M\to T_pM$ is a linear isomorphism. Define the <orientation> of $T_{\widehat p}\widehat M$ to be the one mapped to $o_p$. On $\widehat U_\omega$ it is represented by $\pi^*\omega$. Where two sheets overlap, their base <orientation> forms differ by a positive function, so their pullbacks determine the same <orientation>. This proves the \b[canonical <orientation> of the cover]:
$$
\boxed{o_{\widehat p}=(d\pi_{\widehat p})^{-1}o_p.}
$$
The deck involution $(p,o_p)\mapsto(p,-o_p)$ reverses this <canonical orientation of the orientation double cover>. No global <orientation> of the base was used.
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