In a recent paper, De Stefani and N\'{u}\~{n}ez-Betancourt proved that for a standard-graded $F$-pure $k$-algebra $R$, its diagonal $F$-threshold $c(R)$ is always at least $-a(R)$, where $a(R)$ is the $a$-invariant. In this paper, we establish a refinement of this result in the setting of complete intersection rings.