I ran into a similar problem recently.
Devin's solution, if I'm understanding it correctly, is to present a prompt to whomever picks up first and, if they don't respond within the timeout, to then redial the remaining numbers in the group.
That creates a bad experience for the caller -- they end up waiting longer to connect to a human -- and for the call targets -- who may have their phones ring briefly multiple times.
Being able to present a prompt to everyone dialed and connect whomever responds to the prompt first would solve this problem, but it doesn't appear to be possible to do with the API as it stands, currently.