Since thisCan bounties be awarded after expiration? concluded that this is not really intended behaviour apparently, I propose a period of 24 hours after the bounty has expired during which the question will not be on the featured tab but the promoter can still manually award the bounty. The "cost" of this would be that no new bounty could be set up until the old one were awarded.
Similar to Bounty questions should have a grace period to accept an answer after 7 days, which got closed as a duplicate without duplicate link... (Well, according to Robert's comment there, it's Option to extend bounties by 24 hours., to which he replied with basically what I am proposing here, but an answer is not a feature-request...)