Peano curves with smooth footprints

We construct Peano curves $\gamma : [0,\infty) \to \mathbb{R}^2$ whose "footprints" $\gamma([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus \{\gamma(0)\}$. Moreover, these boundaries can be taken $C^\infty$-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.