Step up key piece of activity after usual close of business? (8)
I believe the answer is:
escalate
'step up' is the definition.
('escalate' can be a synonym of 'step up')
'key piece of activity after usual close of business?' is the wordplay.
'key' becomes 'esc' ('Escape' key on a computer keyboard).
'piece of' indicates taking the first letters.
'after usual close of business?' becomes 'late' (I am not sure about this - if you are sure you should believe this answer much more).
The initial letter of 'activity' is 'a'.
'esc'+'a'+'late'='ESCALATE'
Can you help me to learn more?
(Other definitions for escalate that I've seen before include "rocket" , "Increase in extent or intensity (8)V" , "Increase in intensity or extent, like trouble brewing" , "Cause to increase, say dispute (8)." , "Rapidly increase in intensity" .)