# Is it known which links have Seifert fibered complements?

I believe many such links can be constructed by looking at a foliation similar to the hopf fibration, but the wrapping leaves replaced with $(p,q)$ torus knots. However, I'm interested in particular in whether there are classes of examples that don't fall under that construction.

But you are basically right, the links you get are fibres (possibly degenerate) in some (possibly singular) Seifert fibration of $S^3$ and pretty much it can only be fibred in the ways you mentioned.
The book of David Eisenbud and Walter Neumann on Three-dimensional link theory and invariants of plane curve singularities (Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985) contains a discussion of Seifert links (Chapter II, Section 7). In particular, yes, there are many more Seifert links than $(p,q)$ torus knots or links, for instance, the "key-chain" links. For more details, see also the paper of Ryan Budney, JSJ-decompositions of knot and link complements in $S^3$, Enseign. Math. 52 (2006), no. 3-4, 319–359.