Going out on a limb here...
I see in your sig line that you're playing a Jazz Bass. If this is the bass that the A is jumping out of the nut on, my guess is that 1) your neck is really bowed and needs to be adjusted, which is probably not the case... or 2) the A string was clipped too short and rides up to the top of the post on the A tuner, thereby preventing you from getting good pressure on the nut - which is where I'm putting my money here.
Look at the angle the A string makes as it goes over the nut, and compare it to the other 3 strings. If the A makes little or no angle, and you can only count maybe one or two wraps of string around the peg, then I'd bet dollars to doughnuts that's your issue. Replace the A string, and when you do, clip the string NINE INCHES past the center of the peg. Install carefully, letting each wrap go below the next, like the threads of a screw, finally letting the A string leave the post near the face of the headstock.
The extra wraps of string will help to hold the string down on the post, thereby creating pressure on the A, and preventing it from hopping out.
It's a long post, but I'm punchy and wired after a good wedding gig, so for better or for worse, you get a novella instead of an answer. HTH!
