Lets assume I really like a certain question and I would like to get the best answer possible from the community, but there is already a reward announced by the author. Is there a way to assign additional bounty (of an equal or not equal value) to a question that another person already set a bounty for, in order to promote the question further?
Yes, and no.
While one bounty is already running, you cannot assign another one. You won't find the button for it. Once the existing bounty is awarded or expired, you can set a new bounty for the question. If it is your first bounty and you have not answered the question, you can assign 50 reputation points or more. It is double the amount if you have answered the question, and for each additional bounty. The increments are capped at 500 reputation points.
You can choose to award any answer, if one answer has already received the bounty, the value shown in the blue box will show the sum of all bounties.
You can check the timeline (or the edit history) for whether or not bounties have previously been applied:
For example, the timeline for Theoretical basis behind orbital correlation diagram for pericyclic reactions can be found here. (There are scripts like this on stackapps.com, which can add a link to the timeline for you.)