The search for a one-dimensional (1D) system with purely 1D bands and strong Rashba spin splitting is essential for the realization of Majorana fermions and spin transport but presents a fundamental challenge to date. Herein, using first-principles calculations, we demonstrated that atomic Tellurium (Te) chains exhibit purely 1D bands and giant Rashba spin splitting, and their splitting parameters depend strongly on strain and structure distortion. This phenomenon stems from the helical structure of atomic Te chains, which can not only sustain significant strain but also realize the synergy of orbital angular momentum and in-chain potential gradient in enhancing spin splitting. The structure distortion of stretched helical Te chains is critical to execute this synergy, generating a large Rashba spin splitting among the known systems. Our findings proposed a potential 1D giant Rashba splitting system for exploring spintronics and Majorana fermions, and provide routes for engineering spin splitting in other materials.