In general the answer is "no". For example, consider $K = 4_1$, the figure-eight knot. Then neither its double cover nor its triple cover is a link complements. One of its four-fold covers is a link complement, the other is not. One of its five-fold covers is a link complement, the other three are not.
As we take larger and larger covers, we generically expect torsion to appear in homology, and thus the cover does not embed in $S^3$.
On the other hand, the figure-eight does have infinitely many covers which are link complements. This is because the "good" four-fold cover is the complement of the link $10^2_{138}$ which has an unknotted component.