Rational Fibrations of ${\bar{M}}_{5,1}$ and ${\bar{M}}_{6,1}$

This is the second of two papers on the birational geometry of ${\bar{M}}_{g,1}$. We construct rational maps from ${\bar{M}}_{5,1}$ and ${\bar{M}}_{6,1}$ to lower-dimensional moduli spaces. As a consequence, we identify geometric divisors that generate extremal rays of the effective cones for these spaces.