More etale covers of affine spaces in positive characteristic

We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, etale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H. This improves our earlier result in math.AG/0207150, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.