Well, I don't really know the reason why the stars are where they are. But here's how I remember it:
- You can push-foward the the tangent space for any morphism of smooth manifolds, and you can pull-back the cotangent space.
- The cotangent bundle is usually written using the star for dual as $T^*M$.
- Ergo, pull-backs by smooth maps are $\phi^*$.
Note that it is a common convention that the "upper" starred objects map things in the "reverse" direction. $\phi: M\to N$ and $\phi^*: TN\to TM$ for smooth manifolds. $A: V\to W$ a linear map between inner-product spaces, the adjoint $A^*:W\to V$ goes the other way.
If I were to engage in some baseless speculation:
In the context of differentiable manifolds (where this question seems to be motivated), the induced (by a smooth map between manifolds) mapping of the cotangent bundle is the first one that you really need a new notation for. For the tangent bundle, if $\phi: M\to N$ you can just use $d\phi: TM\to TN$. So to me it is quite plausible that someone needed a notation for the induced mapping of the cotangent bundle, and decided to adorn $\phi$ with a star (for reasons unknown). Perhaps there is already an established tradition of adding this type of symbols in the superscript position (perhaps to allow enumerating several maps $\phi^*_1, \phi^*_2, \ldots$). And when it comes time where a symbol for the pushforward is needed, it is actually quite natural to just move the star down as it were.