The Neutrino Elastic-scattering Observation with NaI(Tl) experiment (NEON) aims to detect coherent elastic neutrino-nucleus scattering (CEνNS) in a NaI(Tl) crystal using reactor anti-electron neutrinos at the Hanbit nuclear power plant complex. A total of 13.3 kg of NaI(Tl) crystals were initially installed in December 2020 at the tendon gallery, 23.7 ± 0.3 m away from the reactor core, which operates at a thermal power of 2.8 GW. Initial engineering operation was performed from May 2021 to March 2022 and observed unexpected photomultiplier-induced noise and a decreased light yield that were caused by leakage of liquid scintillator into the detector due to weakness of detector encapsulation. We upgraded the detector encapsulation design to prevent the leakage of the liquid scintillator. Meanwhile two small-sized detectors were replaced with larger ones resulting in a total mass of 16.7 kg. With this new design implementation, the detector system has been operating stably since April 2022 for over a year without detector gain drop. In this paper, we present an improved crystal encapsulation design and stability of the NEON experiment.