After Simon “[becomes] inarticulate in his effort to express mankind’s essential illness.” (Golding 89) to the other boys, he knows that they will not believe him if he just simply tells them about the Beast, so he knows that the only way they will believe him is if he finds proof.
Unfortunately, Simon fails to successfully use spiritual power to keep the boys from worrying about the beast.
One of the initial problems Simon has is his shyness and stage-fright at the assemblies while trying to tell the boys about the beast.
He also accepts that the boys might not believe him, which directly shows him using spiritual power by standing alone with his beliefs.
Another way Simon uses spiritual power is when he shows innate goodness any way he can.