To perch on or grasp the objective surface using the suction cup-based manipulator, the precise contact control is commonly required. Improper contact angle or insufficient contact force may cause failure. To enhance the tolerance to flight control insufficiency, a suction cup that comprises an inner soft cup and an outer firm cup to facilitate its engagement without reducing the adhesion stiffness is investigated. The soft cup is adaptable to the angular error induced by the multicopter and the resulting adhesion force can draw the firm cup and correct the angular error between the firm cup and the surface. These effects increase the engagement rate and reduce the dependence on precise control. The outer firm cup is devoted to providing a large adhesion force and a stiff base for subsequent tasks. To reduce the air evacuation time in the firm cup, a novel self-sealing structure is designed. Based on the combined cup, we build a multifunctional aerial manipulation system which can execute perching or lateral aerial grasping tasks. With the proposed prototype, the comparative flight experiments involving perching on a wall under disturbance and grasping an object are conducted. The results demonstrate that our proposed suction cup outperforms the conventional cup.