Peano curves with smooth footprints

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

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