Let $S_1,S_2,\ldots S_k$ be a sequence of sets. We will call this sequence expanding if $S_i$ is not covered by $S_1,\ldots S_{i-1}$ i.e. $S_i$ contains at least one new element. Let $C_p$ be a cyclic group of size prime order $p$.

It is easy to show that for every $A\subset C_p$ there exists at least $k=\frac{p}{|A|}$ elements $a_1,\ldots a_k$, such that the sets $A+a_i$ are expanding.(Since each time we cover at most |A| new elements and we can cover all elements)

My question is if it is possible to improve this bound for $A$ of size at most $p/2$ to $k=\frac{p\log|A|}{|A|}$?