Not to get caught up with the thread (6)
I believe the answer is:
cotton
'the thread' is the definition.
(cotton is a kind of thread)
'not to get caught up' is the wordplay.
'get' means one lot of letters go next to another.
'caught' becomes 'c' (cricket abbreviation).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'not'+'to'+'c'='nottoc'
'nottoc' written backwards gives 'COTTON'.
'with' acts as a link.
(Other definitions for cotton that I've seen before include "Soft fabric and fluffy wool (6" , "Source of cloth" , "This wool is a fluffy wadding" , "Plant fibre for fabrics" , "Fabric made from tropical plant - old man river don't plant it" .)