Hilary Stone has some good info about head clip headsets:
Depending on when/where/by whom the frame was made, the headset will have been designed to fit with a 1" (external diameter) steerer or a 31/32" steerer (external again). My Hetchins has a 1" steerer despite being pre-war (31/32 was more common then) so a normal Cinelli 1A fits in fine and securely without needing the expander. I guess you might have a smaller diameter stem installed by someone thinking the steerer was 31/32" when it's actually 1" ?
The fact that it's a tandem headset probably complicates things as well. If the stem will tighten with the expander further down, you could try on of those really long Nitto quill stems to see if the extra length allows you to get it secure and shim it at the clip to stop rattles.