I don't understand why this gets downvoted for merely disliking the suggestion. The downvote button says: "The question does not show any research effort, it is unclear or not useful." It doesn't say: "downvote if you dislike the idea." This is stupid, as it stymies ideas to be put up for discussion, when those who do so must fear to be punished for it if the idea may not be the mainstream view ... The consequence is a self-reinforcing echo chamber in which anything not-aligned is silenced. Personally I won't share any ideas here anymore.
I generally agree that it makes sense to limit the bounty period to 7 days so that the active bounty question section (i.e. "bountied tab") does not get overcrowded. But merely technically speaking, the bounty could still be awarded after the question has been removed from the "bountied tab". I would not see any downside in that. What harm would this do to anybody ?
Yet the rules don't allow a later upgrade:
According to https://superuser.com/help/bounty and my own experience, the full bounty is auto-awarded automatically only if ...
If the bounty was started by the question owner, and the question owner accepts an answer posted during the bounty period, and the bounty expires without an explicit award then we assume the bounty owner liked the answer they accepted and award it the full bounty amount at the time of bounty expiration.
source: https://superuser.com/help/bounty
Case in point: I just missed the 7 day bounty period (lots of things going on in my life) and accepted an answer afterwards, which had already been auto-awarded half the bounty, but this did not double the bounty. The user still got only half the bounty awarded. So my acceptance of the answer did not have an effect on the bounty.
I don't understand what difference it makes if the answer is accepted after 5 days or 9 days ...
Maybe there is a good reason for this rule. But if there is none, I suggest there should still be a way to award the full bounty at any time. Either one of these two implementations would be fine:
Automatically awarding the full bounty by the system whenever the above-cited criteria are met (even after the 7 day period ended).
OR: When the user accepts an answer beyond the 7 days, the grayed out bounty-award button becomes active again, so that the user can manually award the bounty (ONLY to the auto-awarded answer in case it is seen as problematic to switch the bounty as mentioned in the first comment below).
This is not a duplicate of A grace period of one day to award the bounty after expiration (without remaining featured, of course) as (one of my) suggestions is an open-ended possibility to upgrade the auto-awarded bounty to a full bounty (while keeping the 7 days limit for the active bounty tab) ... not a 1-day grace period, which BTW has been implemented already! So this other question, which has been asked more than 13 years ago, is long obsolete.