Anon-fully conjugated polymer as a new class of acceptor materials has shown some advantages over its small molecular counterpart when used in photoactive layers for all-polymer solar cells (all-PSCs), despite a low power conversion efficiency (PCE) caused by its narrow absorption spectra. Herein, a novel non-fully conjugated polymer acceptor PFY-2TS with a low bandgap of similar to 1.40 eV was developed, via polymerizing a large pi-fused small molecule acceptor (SMA) building block (namely YBO) with a non-conjugated thioalkyl linkage. Compared with its precursor YBO, PFY-2TS retains a similar low bandgap but a higher LUMO level. Moreover, compared with the structural analog of YBO-based fully conjugated polymer acceptor PFY-DTC, PFY-2TS shows similar absorption spectrum and electron mobility, but significantly different molecular crystallinity and aggregation properties, which results in optimal blend morphology with a polymer donor PBDB-T and better device physical processes in all-PSCs. As a result, PFY-2TS-based all-PSCs achieved a PCE of 12.31% with a small energy loss of 0.56 eV enabled by the reduced non-radiative energy loss (0.24 eV), which is better than that of 11.08% for the PFY-DTC-based ones. Our work clearly demonstrated that non-fully conjugated polymers as a new class of acceptor materials are very promising for the development of high-performance all-PSCs.