The incompressible Seifert surface $\Sigma$ will have incompressible preimage $S$ in the double branched cover if and only if the complement of a tubular neighborhood of $\Sigma$ in $S^3$ has incompressible boundary. This condition was termed "partially unknotted" by Jaco.

Examples of Seifert surfaces which are *not* partially unknotted were claimed by Jaco in the paper, but not explicitly described.
The simplest examples can be obtained by taking a knotted handlebody with hyperbolic complement, such as Thurston's knotted wye.

See also Adams-Reid for descriptions of knotted genus 2 handlebodies (with incompressible hyperbolic complements).

Now, draw a knot on this handlebody which bounds a genus 1 Seifert surface $\Sigma$, using the fact that for a genus one surface $\Sigma$, $\Sigma\times [0,1]$ is a genus 2 handlebody. Any such knot will have a Seifert surface which is not partially unknotted.