I believe the answer is no. Consider $\phi_0(x)=x^4+x^3$ on the field $F$ of size $q=2^7$. Then $\text{max}_a\lvert\phi_a(F)\rvert=83<2q/3$.

Actually, I believe that for each $\gamma>1-e^{-1}$ (Euler $e=2.71\dots$) there is a map $\phi_0$ on a suitable finite field $F$ of even order $q$ such that $\text{max}_a\lvert\phi_a(F)\rvert\le\gamma q$.

(Initially, 83 was 79, a miscalculation as pointed out by Boris Bukh.)

Comment: Tomorrow I hope to add some more details