Given NFA to recognize a language, construct a new NFA for that language without the empty string
Why go for a subtle or elegant answer? You can drive a screw in with a jackhammer, too.


Define a DFA $M_{\epsilon}^{-1}$ for the language $\Sigma^* \setminus \{\epsilon\}$. Hint: it has two states for any $\Sigma$.
Say how to construct $M'$ such that $L(M') = L(M) \cap L(M_{\epsilon}^{-1})$. Hint: use the Cartesian Product Machine construction.