What makes a language hard in a computational sense is neither simply that it contains very few words(e.g. is finite) or that it contains a lot of words(e.g. is infinte) but rather an intricate selection of a subset of $\Sigma^*$.
I would try to formalize this intuition as the following statement:
Let $A,B$ be two languages which are complete w.r.t to some class $\mathcal{C}$. Then for every language $C$ it holds that $$ A \subseteq C \subseteq B \implies C \text{ is complete w.r.t $\mathcal{C}$} $$
What can we say about the truth value of this statement or interesting special cases such as NP. Maybe it implies only lower bounds(hardness)? I would be rather suprised if even the lower bounds of $A$ and $B$ don't apply to $C$.
One natural example to explore might be HAMCIRC and HAMPATH. If the hypotheses holds then we could state that any set of graphs $G$ for which every element has a hampath and every graph with a hamcirc is contained in it, it follows that $G$ is NP-complete.