Bounty question: improve existing answer or be eligible for bounty?

Question

Imagine this situation: I posted an answer to a question. After some time the OP decided he needs a better answer or more attention and decides to put a bounty on his question.

Now I want to add a - possibly better - different solution to the question. What do I do?

1. Merge it with the existing answer. Then I am not eligible for the bounty (at least not automatically), since the answer was posted before the bounty was in effect.
2. Add another answer. While I would not normally do this, the system allows it. I'll be eligible for the bounty, even when awarded automatically.
3. Delete the existing answer and post a new answer (maybe with the original answer merged in it). I lose all reputation points gained from the deleted post, but am eligible for the bounty.

Answer 1

If your new answer is radically different (as in, it talks of something else and doesn't have the original point) do Option 2 or 3.

Otherwise, just edit it in (Option 1). The automatic award shouldn't be much of a problem. It's only there for when the OP forgets to award the bounty (remember, there's a 24hr grace period for this as well), or when the OP doesn't like any of the answers (in which case you really don't deserve the bounty :P)