We are interested in estimating the size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$. Let $p$ be an odd prime, $g$ a primitive root modulo $p^2$, and $A=\langle g^p\rangle$ the unit subgroup of order $p-1$. Experimentation suggests strongly that if $p \geq 61$, then the size of the sumset $3A = A + A + A$ approximates the order of the ring. This observation is due to Felipe Voloch (see the original question). In particular, we conjecture that $$|3A|\geq p^2-1,$$ for such $p$. It is known that, as sets, $4A = \mathbb{Z}/p^2\mathbb{Z}$. Additionally, numerical evidence suggests that $3A \supseteq (\mathbb{Z}/p^2\mathbb{Z})\setminus\{0\}$, and that $p\equiv 1\pmod{3}$ implies strict containment (i.e., $3A = \mathbb{Z}/p^2\mathbb{Z}$). Perhaps these observations could provide a foothold toward the result. Is there work that addresses this phenomenon, or any existing progress toward an explanation of the estimate above?

(Edit: The remark about "strict containment" refers to the $\supseteq$ relation. That is, $3A \supsetneq (\mathbb{Z}/p^2\mathbb{Z})\setminus\{0\}$ whenever $p\equiv 1\pmod{3}$, which implies that $3A=\mathbb{Z}/p^2\mathbb{Z}$.)