Assessing complexity of a language using subset relations
The statement is unlikely to be true since we can deduce NP-hardness for Graph Isomorphism from it (which would imply a collapse of PH to its second level) as follows:

Let SGI be the Subgraph Isomorphism problem which is known to be NP-complete. Then GI is a subset of SGI. Now, take HAMCIRC and define a slightly different version:

ISO-HAMCIRC = $\{ (G_1, G_2) | G_1 \text{ is isomorph to } G_2 \text{ and } G_1 \in \text{HAMCIRC} \}$ 

It is trivial to see that ISO-HAMCIRC is NP-complete as well. 

It follows that ISO-HAMCIRC $\subseteq$ GI $\subseteq$ SGI and therefore GI would be NP-hard assuming the hypothesis holds.