If every string appears in $f$ consecutively, then every string appears consecutively in $f \sigma$.
(So yes in answer to the title question, no in answer to the question as phrased in the question body :)
To see this, suppose that every string appears consecutively in $f$. Let $S$ be a string. Since $S 0 S$ appears consecutively in $f$, it follows that $S$ appears consecutively in $f$ starting from an even position. (We could equally have used $S1S$.)
Since every string appears consecutively $S$ in $f$ starting from even position, it's also the case that $S \sigma$ appears consecutively in $f$ starting from an even position. So $S$ appears consecutively in $f \sigma$ (starting from even position).