Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ring $R\subsetneq R'\subset \tilde R$ is not reflexive.
Let $m=(t^3,t^4,t^5)$ be that maximal ideal of $R$. As the semigroup generated by $(3,4,5)$ contains all numbers from $3$, $mR'\subseteq m\tilde R\subset R$. It follows that $Hom_R(R',R)$ contains $m$, and since it has to be an ideal of $R$ and can not be $R$, it is $m$.
So now we just need to show that $Hom_R(m,R)\neq R$. But multiplication by $t$ maps $m$ to $R$, and $t$ is not in $R$.
In general, I think for a non-Gorenstein domain of dimension one, torsion free modules are rarely reflexive.