Consecutive non-square non-primitive pairs in a finite field

Let $q$ be an odd prime power and write $${\theta }_q:=\frac{\varphi (q-1)}{q-1}\mathrm{.}$$ If ${\theta }_q<\frac{1}{3}$, or if ${\theta }_q=\frac{1}{3}$ and $q\mathrm{\notin }\{7,13,19,25,37\}$, then the finite field $\text{F}$ contains a pair of consecutive elements that are both non-square and non-primitive. This extends a result of Jarso and Trudgian for prime fields $\text{Fp}$, where the same conclusion was obtained under the stronger condition ${\theta }_p\le \frac{1}{4}$. More generally, let $\mathrm{\ell }$ be the least odd prime divisor of $q-1$. If ${\theta }_q\le \frac{1}{3}$, then $\text{F}$ contains a pair of consecutive elements that are non-squares and $\mathrm{\ell }$th powers, with the sole exceptions $q\in \{7,13,19,25,37,43\}$.