Can you start a bounty on a question that already has an accepted answer, if you feel that your answer to the question is better than the one that was accepted?
Yes, you can. Do note that the minimum bounty will be 100 reputation points (since you've already posted an answer to the question) and you can never award the bounty to your own answer.
Everything you want to know should be covered by the FAQ entry How does the bounty system work? on Meta Stack Exchange.