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 an earlier result of the author, which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point.