Orlicz norms are the general framework used for these sorts of results, though much of the Orlicz norm literature is concerned with concentration of sums $\sum_i X_i$ or suprema $\sup_i X_i$, rather than $\ell_2$ norms. If you want less theoretical literature recommendations, some authors have been looking into "sub-weibull" random variables lately. If one defines $\psi_\alpha(x) = \exp(x^\alpha)-1$, and examines the Orlicz norm $\lVert X\rVert_{\psi_\alpha}$ associated with this Young function, then
- $\alpha = 2$ yields precisely the Sub-gaussian parameter of $X$,
- $\alpha = 1$ yields precisely the sub-exponential parameter.
For $\alpha < 1$ one no longer has that $\lVert \cdot\rVert_{\psi_\alpha}$ is a norm (essentially because the function $\psi_\alpha$ is no longer convex). Still, some can try to recover parts of the general theory, which is precisely what the papers on "sub-weibull" random variables try to do.
Decreasing $\alpha$ is of interest because $\lVert X^2\rVert_{\psi_\alpha} = \lVert X\rVert_{\psi_{\alpha/2}}$, i.e. if one starts with sub-gaussian bounds on $X$, then one gets sub-exponential bounds on $\lVert X\rVert_2$.