Let $X$ be a centered and bounded random variable on $[-a, a]$. By Hoeffding's lemma, $X$ is subgaussian with variance proxy $a^2$. As a reminder, $X$ is subgaussian with variance proxy $\sigma^2$ if $$E[\exp(tX)]\leq \exp(t^2\sigma^2/2) \quad\forall t\in\mathbb{R}.$$
The proof of Hoeffding's lemma, eg here, seems to rely on the variance of $X$ being bounded by $a^2$. This is usually a rather crude bound. If I know the variance of $X$, can I obtain a sharper bound on the variance proxy? Eg, is it true that the variance itself is a variance proxy? A first look at the proof in the link seems to suggest to me that the final bound is simply the variance.
Or perhaps some other nice function of the variance is a variance proxy?
Thanks!