Electron-hole recollisions are induced by resonantly injecting excitons with a near-IR laser at frequency fNIR into quantum wells driven by a 10  kV/cm field oscillating at fTHz=0.57  THz. At T=12  K, up to 18 sidebands are observed at frequencies fsideband=fNIR+2nfTHz, with -8≤2n≤28. Electrons and holes recollide with total kinetic energies up to 57 meV, well above the ELO=36  meV threshold for longitudinal optical (LO) phonon emission. Sidebands with order up to 2n=22 persist up to room temperature. A simple model shows that LO phonon scattering suppresses but does not eliminate sidebands associated with kinetic energies above ELO.