Swarms of self-propelled particles exhibit complex behavior that can arise from simple models, with large changes in swarm behavior resulting from small changes in model parameters. We investigate the steady-state swarms formed by self-propelled Morse particles in three dimensions using molecular dynamics simulations optimized for graphics processing units. We find a variety of swarms of different overall shape assemble spontaneously and that for certain Morse potential parameters at most two competing structures are observed. We report a rich "phase diagram" of athermal swarm structures observed across a broad range of interaction parameters. Unlike the structures formed in equilibrium self-assembly, we find that the probability of forming a self-propelled swarm can be biased by the choice of initial conditions. We investigate how thermal noise influences swarm formation and demonstrate ways it can be exploited to reconfigure one swarm into another. Our findings validate and extend previous observations of self-propelled Morse swarms and highlight open questions for predictive theories of nonequilibrium self-assembly.