  Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let
$\mathcal{P}$ be the class of $p$-groups of order $p^n$ ($n\geq 3$). Consider
the function $\alpha:\mathcal{P}\longrightarrow (0, 1]$ given by
$\alpha(G)=\frac{|C(G)|}{|G|}$. In this paper, we determine the second minimum
value of $\alpha$, as well as the corresponding minimum points. Further, since
the problem of finding the second maximum value of $\alpha$ was completely
solved for $p=2$, we focus on the case of odd primes and we outline a result in
this regard.
