We study the conformational behavior of a polymer adsorbed at an attractive stringlike nanowire and construct the complete structural phase diagram in dependence of the binding strength and effective thickness of the nanowire. For this purpose, Monte Carlo optimization techniques are employed to identify lowest-energy structures for a coarse-grained model of a polymer in contact with the nanowire. Among the representative conformations in the different phases are, for example, compact droplets attached to the wire and also nanotubelike monolayer films wrapping it in a very ordered way. We here systematically analyze low-energy shapes and structural order parameters to elucidate the transitions between the structural phases.