The reason is currently defined as:
Reward existing answer
One or more of the answers is exemplary and worthy of an additional bounty.
This bounty reason is the only one present that currently indicates that an existing answer will be rewarded with the bounty in the end. There have been requests declined to allow these bounties to be rewarded without having to wait 24 hours and even a request to hide these bounties from the featured tab. All the reasons seem to be centered around the idea that other answers that may be posted would not be eligible for the bounty, and that having to wait serves no purpose.
I have frequently expressed my opinion that this is misuse of the bounty reason. It's not intentional misuse, though. It's misused because the reason is poorly defined which encourages the wrong behavior. Following the purpose of bounties, any answer that exists or comes to be throughout the period of the bounty should be eligible. This bounty reason should encourage the holder to award it to answers which may be better than an answer which the holder originally intended to award the bounty.
Here is a better, reworked edition of the bounty reason:
Reward outstanding answer
Any exemplary information that can be provided or already exists may be rewarded.
I believe this reason is general enough to fit in with the other bounties while still encouraging participation from other users. I believe this would take the focus away from just flat-out transferring reputation to a user because they helped them out and put it on the general need for outstanding information.