Character also falling sick after Jack? (4)
I believe the answer is:
jill
'character also falling' is the definition.
(Jack and Jill tumble down in the nursery rhyme)
'sick after jack?' is the wordplay.
'sick' becomes 'ill' (synonyms).
'after' is a charade indicator (letters next to each other).
'jack?' becomes 'J' (playing card abbreviation).
'ill' after 'j' is 'JILL'.
(Other definitions for jill that I've seen before include "Female ferret" , "Jack's pet" , "Jack's nursery rhyme girl companion" .)