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Next-generation particle accelerators, such as the proposed Large Hadron Collider (LHC) energy upgrade (highenergy LHC), will require very-high-field (> 20 T) superconducting magnets. This paper describes the progress toward this goal made to date as a part of a collaborative work between Particle Beam Lasers, Inc., and Brookhaven National Laboratory. To reduce the cost, high-temperature superconductors (HTSs) are used in a hybrid design with conventional low-temperature superconductors (LTSs) Nb3Sn and NbTi. The focus of this paper is on using a second generation (2G) ReBCO HTS tape in cosine-theta coil geometry. The complex ends of the cosine-theta geometry are particularly challenging for a brittle HTS tape. We report the construction and 77-K test results, one with a 4-mm and another with a 12-mm ReBCO tape, neither showing measurable degradation. This paper also presents the first successful use of Kapton CI on an HTS tape, which offers many advantages. Future plans include the construction and 4-K testing of a full cosine-theta HTS coil (first in a stand-alone mode and then in a hybrid structure with LTS coils) and the modeling and measurements of magnetization. This paper is a part of comprehensive research and development toward eventually building a high-field accelerator-quality dipole magnet.