You can always start a bounty on a question that is older than 2 days. It doesn't matter who asked the question and/or whether some answer was already accepted. Of course you need to have enough reputation (75) and no more than maximum number of active bounties (3).
I find it optimal, that only one answer per question can be accepted. It helps people coming with similar problems searching for solutions to quickly spot what they are looking for.
To sum up, you have three options how to reward an answer:
- Accept it (+15 rep., your question, one answer only)
- Upvote (+10 rep., any answers, 200 daily limit for upvoted user)
- Start a bounty (starting at +50 rep.)
It is a perfectly valid reason to start a bounty on a question with accepted answer just to award the author of any answer. There is more information in FAQ, the most important rules:
Questions must be at least 2 days old to be eligible for a bounty.
There can only be 1 active bounty per question at any given time.
Users must have at least 75 reputation to offer a bounty, and may only
have a maximum of 3 active bounties at any given time.
You can even award the same answer repeatedly, or award two or more answers for the same question.
And here is the only space for improvement I personally see, that is, if you would like to reward two answers on the same question, which were both correct and both authors showed lots of effort, you can't give the same award to each one of them. If you give let's say
+100 to the first one, you need to give at least
+200 to the second one. Since every time you start a bounty on a question repeatedly, you need to give at least double the previous amount (
+500 is the limit).