  We construct blow-up solutions of the energy critical wave map equation on
$\mathbb{R}^{2+1}\to \mathcal N$ with polynomial blow-up rate ($t^{-1-\nu}$ for
blow-up at $t=0$) in the case when $\mathcal{N}$ is a surface of revolution.
Here we extend the blow-up range found by C\^arstea ($\nu>\frac 12$) based on
the work by Krieger, Schlag and Tataru to $\nu>0$. This work relies on and
generalizes the recent result of Krieger and the author where the target
manifold is chosen as the standard sphere.
