Warm starting LP solver at non-basic feasible solution
LP solvers use simplex typically, so starting from nonbasic solution doesn't have too much sense. You can try finding a basic solution to $x_h$ by minimizing $||x-x_h||_p$ over your feasible region. I believe it's still an LP for $p=1$ and $p=\infty$. However, there is no guarantee warm start combined with this is faster.