This has been status-declined on Meta SE.
At a cursory glance it would make sense to allow bounties that 'Reward Existing Answer' to be applied immediately. However, there are some concerns that have been raised which has resulted in this being marked status-declined:
- A Bounty's primary effect is to draw attention to the post (the question and any existing answers) as a whole. Awarding the bounty immediately short-circuits that exposure, cutting off community views, potentially better answers and more votes on the posts.
- A few users expressed that they even wait the entire 7-day period to ensure that the exemplary answer they are going to award has the most exposure as possible.
- The way bounties work have been adjusted in the past, to prevent users gaming them.* Removing the 24 hour restriction would provide an un-throttled and mostly invisible path to transfer large amounts of reputation between accounts.
- The bounty system's main purpose is to attract users to post or update their answers. The 'Reward Existing' bounty reason is an outlier in this case - it is the only bounty reason to presume the existence of an already-great-answer that does not require further changes. There's no appetite for coding an exception just for this case.
See mindcorrosive's answer which covers most of the points above, and Jon Ericson's answer, which - on top of everything else - calls for the removal (or re-wording) of the 'Reward Existing Answer' reason entirely.
* Restrictions introduced to prevent abuse include: The limit of 3 active bounties, increasing cost of successive bounties on the same question, and doubling the bounty cost if it's on a post you've answered yourself.