Assuming you are using friction shifters? Are you using an narrow style freewheel, to run 6 speeds in a 120mm frame? Sometimes a chain will be too wide for those and not seat on the center cogs. I've had trouble with Regina narrow sets. But it sounds like you have the right type of chain.
I've never had the opposite problem of too narrow a chain. But since you've replaced everything else I would at least try a chain off something else just to see if anything changes. Unless you have a very large low gear, having the wheel all the way back will not solve anything. Italian race bikes should have the adjuster screws all the way in!