  Elastic scattering off nuclei in target detectors, involving interactions
with dark matter and coherent elastic neutrino nuclear recoil (CE$\nu$NS),
results in the deposition of low energy within the nuclei, dissipating rapidly
through a combination of heat and ionization. The primary energy loss mechanism
for nuclear recoil is heat, leading to consistently smaller measurable
scintillation signals compared to electron recoils of the same energy. The
nuclear recoil quenching factor (QF), representing the ratio of scintillation
light yield produced by nuclear recoil to that of electron recoil at the same
energy, is a critical parameter for understanding dark matter and neutrino
interactions with nuclei. The low energy QF of NaI(Tl) crystals, commonly
employed in dark matter searches and CE$\nu$NS measurements, is of substantial
importance. Previous low energy QF measurements were constrained by
contamination from photomultiplier tube (PMT)-induced noise, resulting in an
observed light yield of approximately 15 photoelectrons per keVee
(kilo-electron-volt electron-equivalent energy) and nuclear recoil energy above
5 keVnr (kilo-electron-volt nuclear recoil energy). Through enhanced crystal
encapsulation, an increased light yield of around 26 photoelectrons per keVee
is achieved. This improvement enables the measurement of the nuclear recoil QF
for sodium nuclei at an energy of 3.8 $\pm$ 0.6 keVnr with a QF of 11.2 $\pm$
1.7%. Furthermore, a reevaluation of previously reported QF results is
conducted, incorporating enhancements in low energy events based on waveform
simulation. The outcomes are generally consistent with various recent QF
measurements for sodium and iodine.
