This paper addresses one of the fundamental questions in the theory of hard-disk packings-how order within a system relates to packing density. The algorithm presented is a seed-based, growth protocol in which new disks are added sequentially to the surface of a growing cluster. The angular position of the new disk is chosen based on the minimization of an objective function designed to control order, as measured by the global bond-orientational order parameter psi(6), which varies between 0 and 1 (with 1 indicating perfect hexagonal close-packed order). Modifying the objective function allows the final packing fraction to be biased while maintaining tight control over psi(6). Inside of the range 0 less than or equal to psi(6)less than or equal to 0.70, the targeted order parameter psi(6) is achieved to within two decimal places of accuracy. Furthermore, it is found that random structures (psi(6)similar to 0.01) can be generated with packing fractions in the range 0.40 less than or equal to eta less than or equal to 0.77. Interestingly, the algorithm can produce nonequilibrium hard-disk configurations that are considerably more disordered than those typical of the equilibrium fluid.