quasi-Banach function spaces are subspace of $L_p$

Question

It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_p[0,1]$ for some $p>0$. However, I am not sure about the case for general quasi-Banach function spaces (which are not rearrangement-invariant). Say, a quasi-Banach lattice on $[0,1]$.