I think it was tissue in anticipation of sweating.
You're claiming Kerry took out the 'tissue' before he needed it, and then never used it? I bet you don't really think that.
Kerry took out some file cards or folded paper, or a palm pilot. He took the object very quickly from his jacket pocket and placed it down on the podium and began to open it, or unfold it, or spread them out a bit. The object was handled gingerly, was probably either beige or white, and did not bend or fold as he handled it/them.