Recall the definition of Hardy hierarchy:

$H_0(n)=n+1\\ H_{\alpha+1}(n)=H_\alpha(n+1)\\ H_\alpha(n)=H_{\alpha[n]}(n)$

Where the last rule applies if $\alpha$ is limit ordinal, and $\alpha[n]$ for $\alpha<\varepsilon_0$ are fundamental sequences using their standard definitions. We can generalize this definition to start with any other increasing function $F(n)$:

$F_0(n)=F(n)\\ F_{\alpha+1}(n)=F_\alpha(n+1)\\ F_\alpha(n)=F_{\alpha[n]}(n)$

This hierarchy has the property that, for each $\alpha<\beta$ we have that $F_\alpha$ is dominated by $F_\beta$.

What I'm interested in are functions which, in a way, lie between $H_\alpha$ for $\alpha<\varepsilon_0$ and $H_{\varepsilon_0}$ itself. To be more specific, I'm looking for a function $F$ which dominates each of $H_\alpha$, and $H_{\varepsilon_0}$ dominates each of $F_\alpha$, for $\alpha<\varepsilon_0$ (similar question for $\varepsilon_0$ replaced with smaller ordinals). I don't know if any functions of the type described above exist, and my question is: are such functions known to exist? I suspect something of this sort could've been investigated by people by Andreas Weiermann, but I haven't found anything of my interest.

Thanks in advance.