The cut locus $C(t)$ of the Jordan curve $L_t$ is a rectifiable tree and one can choose the midpoint $m(t)$ of $C(t)$ is the value of $f$ at $t$, giving a continuous function with image in the interior of the Jordan curve for every $t$.

By compactness one can replace the Jordan curves by a family of polygonal ones. The cut locus $C(t)$ of the Jordan curve $L_t$ is a rectifiable tree and one can choose the midpoint $m(t)$ of $C(t)$ is the value of $f$ at $t$, giving a continuous function with image in the interior of the Jordan curve for every $t$.

The cut locus $C(t)$ of the Jordan curve $L_t$ is a rectifiable tree and one can choose the midpoint $m(t)$ of $C(t)$ is the value of $f$ at $t$, giving a continuous function with image in the interior of the the Jordan curve for every $t$.

The cut locus $C(t)$ of the Jordan curve $L_t$ is a rectifiable tree and one can choose the midpoint $m(t)$ of $C(t)$ is the value of $f$ at $t$, giving a continuous function with image in the interior of the Jordan curve for every $t$.