Smooth approximation to a continuous curve

Question

Let $\gamma: [0,1] \rightarrow M$ be a continuous curve in a smooth manifold $M$. Is there a standard way to approximate $\gamma$ by a smooth curve? My thought was to look at every point $p$ where $\gamma$ is not smooth, consider a coordinate chart $(U, \phi)$ containing $p$ and smoothen $\gamma \cap U$. Can this be made precise?