Injury-time recovery from one’s side? (9)

I believe the answer is:
laterally
'from one's side?' is the definition.
I can't tell whether this defines the answer.
'injury-time recovery' is the wordplay.
'injury time' becomes 'late' (I can't justify this - if you can you should believe this answer much more).
'recovery' becomes 'rally' (rally is a kind of recovery).
'late'+'rally'='LATERALLY'
Can you help me to learn more?
(Other definitions for laterally that I've seen before include "Approaching the side" , "Sideways" .)
