We present a background model for dark matter searches using an array of NaI(Tl) crystals in the COSINE-100 experiment that is located in the Yangyang underground laboratory. The model includes background contributions from both internal and external sources, including cosmogenic radionuclides and surface $^{210}$Pb contamination. To improve the model in the low energy region, with the threshold lowered to 1 keV, we used a depth profile of $^{210}$Pb contamination in the surface of the NaI(Tl) crystals determined in a comparison between measured and simulated spectra. We also considered the effect of the energy scale errors propagated from the statistical uncertainties and the nonlinear detector response at low energies. The 1.7 years COSINE-100 data taken between October 21, 2016 and July 18, 2018 were used for this analysis. The Geant4 toolkit version 10.4.2 was utilized throughout the Monte Carlo simulations for the possible internal and external origins. In particular, the version provides a non-Gaussian peak around 50 keV originating from beta decays of $^{210}$Pb in a good agreement with the measured background. This improved model estimates that the activities of $^{210}$Pb and $^{3}$H are the dominant sources of the backgrounds with an average level of 2.73$pm$0.14 counts/day/keV/kg in the energy region of 1-6 keV, using COSINE-100 data with a total exposure of 97.7 kg$cdot$years.
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