The sources, which generate atom-photon quantum correlations or entanglement based on quantum memory, are basic blocks for building quantum repeaters (QRs). For achieving highly entanglement-generation rates in ensemble-based QRs, spatial-, temporal- and spectral-multimode memories are needed. The previous temporal-multimode memories are based on rephrasing mechanisms in inhomogeneous-broadened media. Here, by applying a train of multi-direction write pulses into a homogeneous-broadened atomic ensemble to induce Duan-Lukin-Cirac-Zoller-like Raman processes, we prepare up to 19 pairs of modes, namely, a spin-wave mode and a photonic time bin. Spin-wave-photon (i.e., atom-photon) entanglement is probabilistically produced in the mode pairs. As a proof-in-principle demonstration, we show that the multiplexed source using all the spin-wave modes and feed-forward-control readout increases spin-wave-photon entanglement generation rate by a factor of $\sim$18.3, compared to non-multiplexed source using the individual modes. The measured Bell parameter for the multiplexed source is $2.30\pm 0.02$ combined with a memory lifetime of $30\mu s$.