Peano curves with smooth footprints

We construct Peano curves γ: [0, ∞) → R² whose “footprints” γ([0, t]) , t > 0, have C^∞ boundaries and are tangent to a common continuous line field on the punctured plane R²\ {γ(0)}. Moreover, these boundaries can be taken C^∞-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.