The electronic dispersions of the quasi-one-dimensional organic conductor TTF-TCNQ are studied by angle-resolved photoelectron spectroscopy (ARPES) with higher angular resolution and accordingly smaller step width than in previous studies. Our experimental results suggest that a refinement of the single-band 1D Hubbard model that includes finite-range interactions is needed to explain these photoemission data. To account for the effects of these finite-range interactions we employ a mobile quantum impurity scheme that describes the scattering of fractionalized particles at energies above the standard Tomonaga-Luttinger liquid limit. Our theoretical predictions agree quantitatively with the location in the (k,ω) plane of the experimentally observed ARPES structures at these higher energies. The nonperturbative microscopic mechanisms that control the spectral properties are found to simplify in terms of the exotic scattering of the charge fractionalized particles. We find that the scattering occurs in the unitary limit of (minus) infinite scattering length, which limit occurs within neutron-neutron interactions in shells of neutron stars and in the scattering of ultracold atoms but not in perturbative electronic condensed-matter systems. Our results provide important physical information on the exotic processes involved in the finite-range electron interactions that control the high-energy spectral properties of TTF-TCNQ. Our results also apply to a wider class of 1D and quasi-1D materials and systems that are of theoretical and potential technological interest.We thank Claus S. Jacobsen for providing the single crystals used in our ARPES studies. J.M.P.C. acknowledges the late Adilet Imambekov for discussions that were helpful in writing this paper. He also would like to thank Boston University's Condensed Matter Theory Visitors Program for support and the hospitality of MIT. J.M.P.C. and T.C. acknowledge the support from Fundacao para a Ciencia e Tecnologia (FCT) through the Grants No. UID/FIS/04650/2013 and No. PTDC/FIS-MAC/29291/2017, J.M.P.C. acknowledges that from the FCT Grants No. SFRH/BSAB/142925/2018 and No. POCI-01-0145-FEDER-028887, and T.C. acknowledges the support from the National Natural Science Foundation of China Grant No. 11650110443
