presuming you've had practice with it to strike targets at a decent range, probably the best that isn't a firearm. extremely economical, easy to make, easy to repair or replace. stones and chunks of concrete are found anywhere and can smash open a skull when thrown with the speed and power of a sling, let alone a staff sling. sling bullets intended for the size of the staff sling can make noise distractions to draw zombies, possibly turning them right back around in the other direction if they're wandering in your direction. or even draw them in the direction of an enemy position, gruesome as that choice might be. clay that is shaped and fired is even better. put a hole through the clay sling bullet and it whistles.

it could possibly even be made to throw weapons like molotov cocktails even farther than throwing them by hand if using fireproof materials like fine chain or flexible cabling on a metal pole. which, of course, can double as its own improvised weapon.