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The late-time growth of single-mode immiscible Rayleigh-Taylor instability is investigated over a comprehensive range of the Reynolds numbers ($1\leq Re \leq 10000$) and Atwood numbers $(0.05 \leq A \leq 0.7)$ using an improved lattice Boltzmann multiphase method. We first reported that the instability with a moderately high Atwood number of 0.7 undergoes a sequence of distinguishing stages at high Reynolds numbers, named as the linear growth, saturated velocity growth, reacceleration and chaotic development stages. The spike and bubble at the secondary stage evolve with the constant velocities and their values agree well with the potential flow theory. Owing to the increasing strengths of the vortices, the spike and bubble are accelerated with velocities exceeding than the asymptotic values and the evolution of the instability enters into the reacceleration stage. Lastly, the curves for the spike and bubble velocities have some fluctuations at the chaotic stage and also a complex interfacial structure with large topological change is observed, while it still preserves the symmetry property. To determine the nature of the late-time growth, we also calculated the spike and bubble growth rates by using five popular statistical methods and two comparative techniques are recommended, resulting in the spike and bubble growth rates of about 0.13 and 0.022, respectively. When the Reynolds number is gradually reduced, some later stages cannot be reached successively and the structure of the evolutional interface becomes relatively smooth. Moreover, we observe that the spike growth rate shows an overall increase with the Atwood number, while it has little influence on the bubble growth rate being approximately 0.0215.