Thread of story about school being evacuated (5)
I believe the answer is:
lisle
'thread' is the definition.
(lisle is a kind of thread)
'story about school being evacuated' is the wordplay.
'story' becomes 'lie' (I've seen this before).
'about' is an insertion indicator.
'being evacuated' suggests removing the centre.
'school' with its centre removed is 'sl'.
'lie' enclosing 'sl' is 'LISLE'.
'of' is the link.
(Other definitions for lisle that I've seen before include "Strong cotton fabric" , "Fine smooth cotton thread for stockings" , "(Fabric woven with) a fine cotton thread" , "Maker of hose" , "Strong twisted cotton thread for the strange Ellis" .)