Bredon [Bre72,Theorem II.6.2] shows that for any Hausdorff space $X$ with the action of a compact Lie group $K$, the natural map $X \to X/K$ has the path-lifting property.
This follows since there is a slice at each point. Bredon attributes this to Montgomery and Yang (The existence of a slice. Ann. of Math. 65 (1957), 108-116).
The general philosophy is that whenever there are slices, there is path-lifting.
Addendum (after it was pointed out I misread the question):
So the above result applies to the situation when $X$ is a manifold and $K$ is acting smoothly and freely to produce a lifted path. But that path may not be smooth a priori.
However, unless I am missing something (I often am, and so please forgive me for that), the proof (I just re-read) in Bredon can be adapted line-by-line to show the resulting path can be taken to be smooth if $K$ acts smoothly and freely on a smooth manifold $X$.
Let $f:I\to X/K$ be a smooth path. $X/K$ is smooth since $K$ is acting freely and properly (since $K$ is compact). Let $\pi:X\to X/K$ be the projection (which is smooth). There are smooth slices (Koszul, 1953) and so there exists a smooth slice $\sigma:I\to f^*X$, which gives your desired smooth path through the pull-back diagram.
[Bre72] Bredon, Glen E. Introduction to compact transformation groups. Pure and Applied Mathematics, 46. Academic Press, New York, 1972.