If it has been sub-categorized by the sub-category-minded, it has so far managed to elude TVTropes as anything more specific than an overlap of I Have Your Wife/Daughter/etc.:
The Big Bad tries to get the hero (or someone else) to do his bidding
by taking someone precious from him and essentially holding them to
ransom. It's often a significant other, frequently a wife or
girlfriend; but any family or friend works[.]
and Trap Is The Only Option:
When the heroes consciously recognize a trap when they see one, but elect to walk into it anyway.
The reasons for this vary. Sometimes, it's because one of the hero's True Companions or a Love Interest is being held hostage, and the hero can't turn his back on them.
, with a dash of He/She Will Come For Me/Us:
A Rescue Trope with the Distressed Damsel/Dude invoking the likelihood of rescue by a specific person or small group. She Will Come For Me is the most basic version, followed by variations. The Distressed Damsel/Dude has usually been kidnapped . . .
Anne M Leone references James Scott Bell, but it's a somewhat vague list for the sake of plot-writing, demonstrating that the notion of composition for a reader doesn't lend itself to snappy trope names:
Bell has created a list of nine different types of cliffhanger endings (or as he calls them, Read On Prompts):
. . .
Announcement of a shattering event
. . .
Question left in the air
i.e., fear for the family's safety / the moment of the kidnapping revealed / will the hero(ine) be able to save them?