yes, when the magnet starts to move away from the loop, the direction of the current in the loop will change direction.
The way I like to think about lenz's law is by thinking about loops of wire hating to have their magnetic field changed, and they're going to do anything they can to avoid it happening. So when you come along with a magnet, and shove it near the end of a wire loops, you're trying to change the magnetic field through the loop from 0 to, say, some positive number (with respect to a chosen axis). The loop hates this, and so acts to generate a field to cancel this out - i.e. a negative magnetic field (with respect to the chosen axis).
So if you had the magnet on the left side of a loop, with the north pole end closest to the loop (i.e. field going from left to right), then the loop is going to generate a field going from right to left.
But when you start to move the magnet away from the loop, you're now trying to decrease the field in the loop, which the loop also hates. So now it's going to try and reinforce the field which you are taking away from it, so the current direction will flip and the field generated will go from left to right.
In your case, you are passing the magnet through the loop, I think. So what happens is the magnetic field is going up and up in the loop, so the induced field is going up and up in the negative direction. But when the magnet is half way, the field is at a maximum in the loop, and so the induced field is zero (when at a maximum, there is no local change), so no current. And when the magnet continues to move out of the loop, the current now flows in the opposite direction. I.e. the current increases in one direction, slows down to zero, and then flows in the other direction, and then slows down back to zero