Let F be a finite field and let E be an n-degree extension of F. Given a family \(\{F_1, \ldots , F_k\}\) of intermediate fields, we discuss the existence of primitive elements of E whose traces over the fields \(F_i\) are prescribed. In 2022, S. Ribas and the author studied this problem and, by employing a very standard approach, we provided asymptotic results under some mild restrictions on the intermediate fields. Moreover, we observed that such element can never exist if \([E:F_i]=2\) for some \(1\le i\le k\) and the corresponding prescribed trace is zero. In this paper we show that, up to this genuine exception, such element exists if n is fixed and \(\# F\) is large enough. In contrast to the ideas employed in our previous work, here our approach basically relies on showing that the corresponding set of elements in E with prescribed traces comprises an affine space with a generic algebraic property. The affine spaces satisfying this property were recently studied by the author, where it is shown that they present a good cancellation through multiplicative character sums (hence they contain a large number of primitive elements).