Consider an unbounded operator \(T\) on a Hilbert space \(\mathcal{H}\). The authors introduce a new type of essential numerical range for \(T\), called \(W_{e5}(T)\) (essential numerical range of type 5). They show, for instance, that \(W_{e5}(T)\) is closed, convex, and contains the essential spectrum \(\sigma_e(T)\). It is also proved that \(W_{e5}(T)\subseteq W_e(T)\), where \(W_e(T)\) denotes the usual essential numerical range of \(T\), and that the inclusion may be strict. Moreover, the authors introduce a notion of \(C\)-numerical range \(W_C(T)\), where \(T\) is assumed to be densely defined and closed and \(C\) is a compact operator on \(\mathcal{H}\). A relation between \(W_{e5}(T)\) and \(W_C(T)\) for finite-rank operators \(C\) is established.