Woman picks engagement rings up (5)
I believe the answer is:
agnes
'woman' is the definition.
'picks engagement rings up' is the wordplay.
'rings' indicates a hidden word (in the sense of encircling/enclosing).
'up' is a reversal indicator (in down clue: letters go upwards).
'senga' is hidden within 'picks engagement'.
'senga' in reverse letter order is 'AGNES'.
(Other definitions for agnes that I've seen before include "Lady" , "girl" , "Scandinavian singer" , "lass" , "Patron saint of virgins" .)