If I'm not mistaken, the Reeb stability theorem prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. Feuilletages, etudies geometriques, Théorème 3.1:

Theorem (Reeb global stability). Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres); the local stability also ensures that the orientation is preserved locally.

I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.

If I'm not mistaken, the Reeb stability theorem prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. Feuilletages: études géométriques, Théorème 3.1:

Theorem (Reeb global stability). Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres); the local stability also ensures that the orientation is preserved locally.

I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.

If I'm not mistaken, the Reeb stability theorem prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. Feuilletages, etudies geometriques, Théorème 3.1:

Theorem (Reeb global stability). Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres).

I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.

If I'm not mistaken, the Reeb stability theorem prevents this from happening for $n\geqslant 3$. There is the following version of it, contained in: C. Godbillon. Feuilletages, etudies geometriques, Théorème 3.1:

Theorem (Reeb global stability). Let $\mathcal F$ be a codimension $1$ foliation of a compact connected manifold; if the boundary of $M$ is non-empty, let $\mathcal F$ be transverse or tangent to the boundary. If $\mathcal F$ has a compact leaf with finite fundamental group, then all the leaves are compact and have finite fundamental group.

and there is a local version saying that such a compact leaf with finite fundamental group is stable. These together imply that the leaves of $\mathcal F$ between two spheres $S_1$ and $S_2$ are all spheres (the global version ensures compactness, and the local version says that locally they are all covering spaces of spheres and therefore spheres); the local stability also ensures that the orientation is preserved locally.

I guess, the key difference from the case of circles is that the circle has infinite fundamental group, so one cannot use Reeb stability in that case.