Geometrization of 3-manifolds has three levels.

Level 1: the compression-body decomposition. This is non-trivial only for the unknot complement, which becomes empty after this decomposition, as the unknot complement is a solid-torus.

Level 2: the connect-sum decomposition. All knot exteriors are prime so this is trivial.

Level 3: the torus decomposition. Satellite knots have this. Typically this is split into the most elementary kind "composite knots" and the more complicated satellite operations, such as Whitehead doubling (hyperbolic types) and cabling (Seifert satellite operations. The hyperbolic satellite operations have an infinite variety.

Seifert-fibred knot exteriors admit two geometries. Non-compact manifolds have a little bit of ambiguity when it comes to putting geometric structures on them. Bonahon's survey of **geometric structures on 3-manifolds** is a good resource for this. In short, you get both a $\mathbb R \times H^2$ and $PSL_2 \mathbb R$ structure. The $\mathbb R \times H^2$ structure is made explicit in my paper via the Milnor Fibration. Moreover, these geometric structures are not rigid in general.

This happens in satellite operations as well. The manifold that comes up via the connect-sum operation is a (trivial) punctured disc bundle over the circle. So there is a full moduli space of such $\mathbb R \times H^2$ structures.

But in short there is (1) the empty set after the compression body decomposition of the unknot exterior (2) hyperbolic knots, but there are also certain hyperbolic links that appear after decomposing satellite knots -- the exact class is all hyperbolic links of $n+1$ components that possess an $n$-component trivial sub-link and (3) the Seifert-fibred knot and link exteriors. These admit related $PSL_2 \mathbb R$ and $\mathbb R \times H^2$ structures and appear in families. The Seifert-fibred knots are the torus knots. The Seifert-fibred links that appear are the "keychain link" that generates connect-sum, and also the links I call "Seifert links" that generate the cabling operation.