We consider a model for a directed polymer in a random environment defined on a hierarchical diamond lattice in which i.i.d. random variables are attached to the lattice bonds. Our focus is on scaling schemes in which a size parameter $n$, counting the number of hierarchical layers of the system, becomes large as the inverse temperature $β$ vanishes. When $β$ has the form $\widehatβ/\sqrt{n}$ for a parameter $\widehatβ>0$, we show that there is a cutoff value $0 < κ< \infty$ such that as $n \to \infty$ the variance of the normalized partition function tends to zero for $\widehatβ\leq κ$ and grows without bound for $\widehatβ > κ$. We obtain a more refined description of the border between these two regimes by setting the inverse temperature to $κ/\sqrt{n} + α_n$ where $0 < α_n \ll 1/\sqrt{n}$ and analyzing the asymptotic behavior of the variance. We show that when $α_n = α(\log n-\log \log n)/n^{3/2}$ (with a small modification to deal with non-zero third moment) there is a similar cutoff value $η$ for the parameter $α$ such that when $α< η$ the variance goes to zero and grows without bound when $α> η$. Extending the analysis yet again by probing around the inverse temperature $κ/\sqrt{n} + η(\log n-\log \log n)/n^{3/2}$ we find an infinite sequence of nested critical points for the variance behavior of the normalized partition function. In the subcritical cases $\widehatβ \leq κ$ and $α\leq η$ this analysis is extended to a central limit theorem result for the fluctuations of the normalized partition function.