Probability bound of the form $P(|X|>\sigma) \geq p$ for sub-gaussian $X$

Question

For $X \sim N(0,\sigma^2)$, I can write the following inequality for any $\mu \in \mathbb{R}$

\begin{align} \mathbb{P}(|X+\mu|> \sigma) \geq \mathbb{P}(|X|> \sigma) \geq \frac{3}{10}. \end{align}

Can we write a similar bound if $X$ is a zero-mean sub-gaussian random variable with variance $\sigma^2$?