1. In baseball, to put all of one's power into one's swing while batting so as to hit a home run. A less common variant of "swing for the fences." 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. I picked up the bat, imagined myself in Wrigley Field, and got ready to swing for the bleachers.

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 impressed when he came out swinging for the bleachers on day one of the trial. After years of middling sales, it's clear that the company is finally swinging for the bleachers with its aggressive new marketing push.