For a finite field F_q^r with fixed q and r sufficiently large, we prove the existence of a primitive element outside of a set of r many affine hyperplanes for q=4 and q=5. This complements earlier results by Fernandes and Reis for q≥7. For q=3 the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For q=2 the set consists only of a single element, and such a result is thus not possible.