Larger hailstones are heavy, and need to be supported by stronger updrafts. So, factors which support strong updrafts will lead to large hailstones. Having lots of CAPE is always a good thing, but it is not necessarily the determining factor of whether a storm produces large hail or not. The likely most important factor is lapse rates, or how much the temperature is decreasing with height in the atmosphere. Steeper lapse rates will lead to lifted air parcels being much warmer than their environment, meaning there will be strong updrafts as those air parcels are very buoyant. Thunderstorms forming in an environment with steep lapse rates (greater than 8 degrees Celsius per kilometer) will be most supportive of the formation of large hail.
Stronger updrafts are also supported by a discrete convective mode. The rotating updrafts of supercells are dynamically stronger than the non-rotating updrafts found in linear convection. Therefore, supercell type thunderstorms will support the formation of large hailstones more than squall line thunderstorms.