1. baseball Literally, to put all one's power into one's swing while batting so as to try to hit a home run. All they need is two more runs to win the game, so you can bet their star batter will come out swinging for the fences.

2. By extension, to put forward one's maximum amount of effort or energy (into or toward something); to act or perform with great intensity or effort. I wasn't sure about their state-appointed lawyer at first, but I was well impressed when he came out swinging for the fences on day one of the trial.