Let $\gamma:[0,1]\to \mathbb{R}^2$ be a rectifiable curve and $\Gamma=\gamma[0,1]$ be its image. Is it possible to cover $\Gamma$ by a countable collection of sets $N,R_1,R_2,\dots$ such that $N$ has vanishing 1-dimensional Hausdorff measure and each $R_i$ admits a bi-Lipschitz embedding into $\mathbb R$?

If yes, is there an analogue of this statement for $k$-rectifiable sets in $\mathbb{R}^n$?