In the field of quantum computation, the non-stabilizerness of a quantum circuit is crucial for understanding and quantifying quantum speed-up. In this work, we explore some intriguing phenomena regarding the non-stabilizerness of a circuit when a Quantum SWITCH structure is employed. This structure is a novel quantum construct that enables quantum states to pass through operations in a superposition of different orders and has shown superiority in numerous tasks over circuits with a definite causal order. Firstly, we discover that the completely stabilizer-preserving operations, which cannot generate magic states under standard conditions, can be transformed into a resourceful operation capable of generating magic states when processed by the Quantum SWITCH. Secondly, when considering the effects of noisy channels on operations, we observe that while the non-stabilizerness of each path may be annihilated, their superposition could still preserve the non-stabilizerness of the operation. These findings reveal unique properties brought by the Quantum SWITCH and open further avenues in future research on magic resources of general quantum architecture.