The object of the present paper is to characterize $K$-contact Einstein manifolds satisfying the curvature condition $R\cdot C=Q(S,C),$ where $C$ is the conformal curvature tensor and $R$ the Riemannian curvature tensor. Next we study $K$-contact Einstein manifolds satisfying the curvature conditions $C\cdot S=0$ and $S\cdot C=0$, where $S$ is the Ricci tensor. Finally, we consider $K$-contact Einstein manifolds satisfying the curvature condition $Z\cdot C=0$, where $Z$ is the concircular curvature tensor.