To increase the practical usefulness of solid-state sensitized upconversion (UC) materials as components of solar energy harvesting systems, it is important to identify and suppress loss mechanisms, and increase the UC quantum yield (Φ UC). Here we focus on a benchmark UC system consisting of the emitter 9,10-diphenylanthracene (DPA) and the sensitizer platinum octaethylporphyrin (PtOEP) in a rigid poly(methyl methacrylate) (PMMA) matrix, and show that one of the major losses originates from Förster resonant energy transfer (FRET) from DPA back to PtOEP. Even though DPA emission lies within the PtOEP transparency window, the quantitative assessment of singlet exciton diffusion for samples with a high DPA content evidences that long-range FRET results in effective exciton trapping by PtOEP. A dramatic factor-of-6 reduction of the DPA emission quantum yield occurs even at PtOEP concentrations as low as 0.05 wt%. To alleviate this problem, we demonstrate a new concept based on the introduction of highly emissive sink sites to trap the singlet excitons produced upon annihilation prior to their quenching by the sensitizer. For DPA/PtOEP blends in PMMA, 1,6-bis-[2,5-di(dodecyloxyphenyl)ethynyl]pyrene is shown to be a useful sink, which results in 1.5-fold increase of the Φ UC. A maximum Φ UC of 2.7% was achieved, which is among the highest reported values for rigid sensitized UC polymers.