I think you are saying that the wire is wound on the larger tube sort of like thread is wound on its spool. That makes for a larger OD and so might not be able to pass thru the toroid hole. I have some toroid transformers about 1/4 inch OD with hole of about
3/32 inches; as a guess, #24 wire was used to wind them. The wire size certainly reduces the space available for any kind of shuttle by a goodly factor. Hence that is why i mentioned the use of a needle or doing the winding by hand.