I have a question with an accepted answer. Recently, I found another solution to the question that works better than the one I accepted.
Is it acceptable to post this as my own answer and re-accept the question?
Yes, it is totally acceptable to do that.
You are encouraged to answer your own questions (there's even a badge for that), and of course you should always accept the answer that worked the best for you. If you truly believe that your answer is the best, then not only is it acceptable to accept your own answer, but that is what you should do.