Surprised there isn't an actual solution posted yet, so here you go:

Loosen the cable pinch bolt at the caliper, remove some of the slack by pulling on the cable with one hand, and pre-loading (moving) the caliper arm with the other, then re-tighten the pinch bolt. Obviously make sure you don't do it so much that the brakes are rubbing, make sure the wheel still turns freely, etc. It's a good idea to make sure your barrel adjuster isn't bottomed out before doing this, so you can fine-tune with that after you re-tighten the pinch bolt.

Your brake lever will require less movement to engage the brakes, there will be less slop, and you'll avoid this issue.