Your only true option is to get new brake shoes for the hub or a new cone as well. Depending on make, it's the shoes or clutch cone that wears first. I assume this is an older hub and not brand new. If it's brand new, the only true option is to change your gearing.

NIB parts are hard to find, but I recommend a suntour coaster brake with shimano coming a close second. Their engagement is the fastest. The engagement is determined by the worm gear ratio.

As a coaster ages, the cone and shoes wear and you can find the engagement not working. You spin a bit then it catches. It sucks. So having new parts in there tightens it up. Also, welding the cog to the driver and welding the cone to the arm will reduce any play. Using a splined front sprocket (bmx style) or a sprocket that has direct mating to the crank (like most road cranks) is the way to go.