  We study an example of a {\em hit-and-run} random walk on the symmetric group
$\mathbf S_n$. Our starting point is the well understood {\em top-to-random}
shuffle. In the hit-and-run version, at each {\em single step}, after picking
the point of insertion, $j$, uniformly at random in $\{1,\dots,n\}$, the top
card is inserted in the $j$-th position $k$ times in a row where $k$ is uniform
in $\{0,1,\dots,j-1\}$. The question is, does this accelerate mixing
significantly or not? We show that, in $L^2$ and sup-norm, this accelerates
mixing at most by a constant factor (independent of $n$). Analyzing this
problem in total variation is an interesting open question.
