How are the hybrid orbitals of sulfur hexafluoride shaped? On the fluorine end the fluoride atoms are simply completing a $3p$ orbital.

On the sulfur end one could posit a hybrid of one $3s$ orbital, two $3p$ orbitals and a three inner $2p$ orbitals. However, I think this would result in a geometry which sulfur hexafluoride does not exhibit.

The explanation I read online is that the orbitals are a hybrid of an $s$ orbital, three $p$ orbitals and two $d$ orbitals.

This confuses me because the $3d$ orbital is quite high energy and I don't think it is possible to have a $2d$ orbital even in a hybrid.

The ground state of a neutral sulfur atom is:

$$ [\ce{Ne}] \underset{3s}{[\uparrow \downarrow]} \underset{3p}{[\uparrow \downarrow \vert \uparrow \vert \uparrow]}$$

With this hybrid the state would be something like the following but with the electrons paired up with the orbitals they are bonded with (also the orbitals are hybridized):

$$ [\ce{Ne}] \underset{3s}{[\uparrow]} \underset{3p}{[\uparrow \vert \uparrow \vert \uparrow]} \underset{3d}{[\uparrow \vert \uparrow \vert \; \; \vert \;\;\vert \;\;]}$$

One approach is that sulfur hexaflouride could be thought of as a less extreme case of $\ce{S^{+6} + 6F^-}$.

Another approach is that sulfur hexaflouride could be a hybrid of a bunch of $\ce{2F2 + SF2}$ arrangements. The $\ce{SF2}$ part has a tetrahedral geometry and so $sp^3$ character. The bonds are then $sp^3-p$ bonds. The total result would be a bunch of partial $sp^3-p$ bonds between the center sulfur and outer flouride atoms and some outer partial $p^3$ bonds between flouride atoms. But I'm not sure what the actual geometry this interpretation would predict or how it would differ in properties.

How are the hybrid orbitals of sulfur hexaflouride shaped?
If you limit consideration to hybridization of atomic orbitals, a good reference to see is On the role of d orbitals in sulfur hexafluoride J. Am. Chem. Soc., 1986, 108 (13), pp 3586–3593.

First, it is found that of the 6 valence electrons that atomic sulfur has (two 3s and four 3p), in SF6 a total of 3.1 electrons worth of electron probability density remain in sulfur atomic orbitals.  

Of the 3.1 electrons:

0.97 are in the 3s orbital

1.82 are in 3p orbitals  

0.24 are in 3d orbitals

and small fractions are in 4s, 4p and 4 d orbitals.  

For a more recent study, see Theory of Hypervalency: Recoupled Pair Bonding in SFn (n = 1−6) J. Phys. Chem. A, 2009, 113 (27), pp 7915–7926