Coordination polymers (CPs) of metal ions are central to a large variety of applications, such as catalysis and separations. These polymers frequently occur as amorphous solids that segregate from solution. The structural aspects of this segregation remain elusive due to the dearth of the spectroscopic techniques and computational approaches suitable for probing such systems. Therefore, there is a lacking of understanding of how the molecular building blocks give rise to the mesoscale architectures that characterize CP materials. In this study we revisit a CP phase formed in the extraction of trivalent lanthanide ions by diesters of the phosphoric acid, such as the bis(2-ethylhexyl)phosphoric acid (HDEHP). This is a well-known system with practical importance in strategic metals refining and nuclear fuel reprocessing. A CP phase, referred to as a "third phase", has been known to form in these systems for half a century, yet the structure of the amorphous solid is still a point of contention, illustrating the difficulties faced in characterizing such materials. In this study, we follow a deductive approach to solving the molecular structure of amorphous CP phases, using semiempirical calculations to set up an array of physically plausible models and then deploying a suite of experimental techniques, including optical, magnetic resonance, and X-ray spectroscopies, to consecutively eliminate all but one model. We demonstrate that the "third phase" consists of hexagonally packed linear chains in which the lanthanide ions are connected by three O-P-O bridges, with the modifying groups protruding outward, as in a bottlebrush. The tendency to yield linear polynuclear oligomers that is apparent in this system may also be present in other systems yielding the "third phase", demonstrating how molecular geometry directs polymeric assembly in hybrid materials. We show that the packing of bridging molecules is central to directing the structure of CP phases and that by manipulating the steric requirements of ancillary groups one can control the structure of the assembly.