Sideways, last-minute improvement? (9)

I believe the answer is:

laterally

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'sideways' is the definition.
(I've seen this before)

'last-minute improvement?' is the wordplay.
'last' becomes 'late' (similar in meaning).
'minute improvement?' becomes 'rally' (recovery after illness or decline. I am not sure about the 'minute' bit.).
'late'+'rally'='LATERALLY'

Can you help me to learn more?

(Other definitions for laterally that I've seen before include "Approaching the side" , "Sideways" .)

I've seen this clue in The Telegraph.

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