Let $X$ be a binomial random variable, $X \sim \mathcal{B}(n,p)$.

When $k > \mathbb{E}[X] = np$, are there no Hoeffding-like bounds on the probability $\mathbb{P}[X \leq k]$?

When $k \leq \mathbb{E}[X]$, it is easy to find that $\mathbb{P}[X \leq k] \leq \exp\{-2n(p-k/n)^2\}$, by means of the Hoeffding bound. When $k > \mathbb{E}[X]$, the Hoeffding bound no longer applies (as far as I can understand), essentially beacuse this would mean taking $t<0$ in the usual formulation of the bound, which is not allowed.

I understand that Markov's bound should still apply, and that $\mathbb{P}[X \leq k] \equiv \sum_{j=0}^k \binom{n}{j} p^j (1-p)^{n-j}$. I also note that when $p=1/2$, the distribution is symmetric, but this is not the case for general $p$. This said, are there no tighter bounds?