Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along $c$ leaves $K$ fibred with $F$ still being the fiber surface?
Such examples were constructed by Morton. He showed that one can find unknotted curves lying on fiber surfaces with zero framing. Twisting about them preserves fiberedness and the fact that it is a knot in $S^3$. Also, curves on the fiber can be pushed disjoint from the fiber meeting your requirement.
Edit: As Danny points out in the comments John Stallings originally came up with this construction.