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NaI(Tl) scintillation materials are considered to be one of many materials that are used exclusively for γ-ray detection and spectroscopy. The gamma-ray spectrometer is not an easy-to-use device, and the accuracy of the numerical values must be carefully checked based on the rules of the calibration technique. Therefore, accurate information about the detection system and its effectiveness is of greater importance. The purpose of this study is to estimate, using an analytical-numerical formula (ANF), the purely geometric solid angle, geometric efficiency, and total efficiency of a cylindrical NaI(Tl) γ-ray detector with a side-through hole. This type of detector is ideal for scanning fuel rods and pipelines, as well as for performing radio-immunoassays. The study included the calculation of the complex solid angle, in combination with the use of various points like gamma sources, located axially and non-axially inside the through detector side hole, which can be applied in a hypothetical method for calibrating the facility. An extended γ-ray energy range, the detector, source dimensions, “source-to-detector” geometry inside the side-through hole, path lengths of γ-quanta photons crossing the facility, besides the photon average path length inside the detector medium itself, were studied and considered. This study is very important for an expanded future article where the radioactive point source can be replaced by a volume source located inside the side-trough hole of the detector, or by a radioactive pipeline passing through the well. The results provide a good and useful approach to a new generation of detectors that can be used for low-level radiation that needs to be measured efficiently.