We study phase-separation behavior of an off-symmetric fluid mixture induced by a "double temperature quench." We first quench a system into the unstable region. After a large phase-separated structure is formed, we again quench the system more deeply and follow the pattern-evolution process. The second quench makes the domains formed by the first quench unstable and leads to double phase separation; that is, small droplets are formed inside the large domains created by the first quench. The complex coarsening behavior of this hierarchic structure having two characteristic length scales is studied in detail by using the digital image analysis. We find three distinct time regimes in the time evolution of the structure factor of the system. In the first regime, small droplets coarsen with time inside large domains. There a large domain containing small droplets in it can be regarded as an isolated system. Later, however, the coarsening of small droplets stops when they start to interact via diffusion with the large domain containing them. Finally, small droplets disappear due to the Lifshitz-Slyozov mechanism. Thus the observed behavior can be explained by the crossover of the nature of a large domain from the isolated to the open system; this is a direct consequence of the existence of the two characteristic length scales.