I found a hack which one can use to ask a question and offer a bounty for it as it becomes visible:
- Compose a new question
- Delete it
- Undelete it after two days or more
- Offer a bounty
Edit3 - Abuse scheme:
As Joshua Dwire pointed out in his answer: I can compose a dummy questions and delete it. Whenever I'd like to pose some real question and offer a bounty immediately - I'll just undelete it, change it and offer a bounty.
Bounties are allowed to be be offered only 48 hours after a question was published (see here), but I believe that they should be allowed to be offered 48 hours after a question becomes visible.
I believe that bounty eligibility should be reset after undeletion.