My guess is that the clamp uses a split collet. The missing bit is a kind of sleeve that fits under the head of the clamp bolt, like a long cylindrical washer. It would probably have an angle sliced off one end (ideally, shaped to fit the column) to match the "nut" which is presumably in there. The clamp bolt pinches the two bits together to grab the column. Done properly, this is a good way to arrange a clamp as it will generally clamp securely without moving or damaging anything. The collets (nut and sleeve bits) are often brass or similar to help avoid scarring the column and they float in their bore. It seems highly likely that the "sleeve" bit dropped out and was lost in the disassembly and move.

But that's all based on one picture that doesn't really show the angles too well! So forgive me if I've misread it.