Disk amplifier design for inertial fusion lasers has evolved with changing fusion-driver requirements from a primary emphasis on gain to a primary emphasis on efficiency. In this paper we compare Shiva and Nova amplifiers to a developmental amplifier (SSA) and show greater than a two-fold improvement in efficiency over past designs under all operating conditions. Experiments to optimize the efficiency of the SSA show that preionization of the flashlamps produces significant benefits and that the packing fraction of lamps is more important than the flashlamp reflector shape. They also show that the optimized flashlamp pulselength and reflector geometry depend on the desired stored energy in the laser medium. We have demonstrated a 7% storage efficiency at a stored fluence per disk of 0.5 J/cm2 (stored energy density of 0.06 J/cm3) and 4% at 2.0 J/cm2 (0.25 J/cm3). Comparison of SSA measurements with storage-efficiency calculations show that our flashlamp model accurately predicts the single-pass pumping of disk amplifiers.