  Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a
prime. For a set $A \subset \mathbb{F}_q$, under certain structural
restrictions, we prove a new explicit lower bound on the size of the product
set $A(A + 1)$. Our result improves on the previous best known bound due to
Zhelezov and holds under more relaxed restrictions.
