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We study the effect of Haldane flux in the bilayer $\alpha$-$\mathcal{T}_3$lattice system, considering possible non-equivalent, commensurate stackingconfigurations with a tight-binding formalism. The bilayer$\alpha$-$\mathcal{T}_3$ lattice comprises six sublattices in a unit cell, andits spectrum consists of six bands. In the absence of Haldane flux, threefoldband crossings occur at the two Dirac points for both valence and conductionbands. The introduction of Haldane flux in a cyclically stacked bilayer$\alpha$-$\mathcal{T}_3$ lattice system separates all six bands, including twolow-energy, corrugated nearly flat bands, and assigns non-zero Chern numbers toeach band, rendering the system topological. We demonstrate that thetopological evolution can be induced by modifying the hopping strength betweensublattices with the scaling parameter $\alpha$ in each layer. In the dicelattice limit ($\alpha = 1$) of the Chern-insulating phase, the Chern numbersof the three pairs of bands, from low energy to higher energies, are $\pm 2$,$\pm 3$, and $\pm 1$. Interestingly, a continuous change in the parameter$\alpha$ triggers a topological phase transition through band crossings betweenthe two lower energy bands. These crossings occur at different values for theconduction and valence bands and depend further on the next nearest neighbor(NNN) hopping strength. At the transition point, the Chern numbers of the twolower conduction and valence bands change discontinuously from $\pm 2$ to $\pm5$ and $\pm 3$ to $0$, respectively, while leaving the Chern number of thethird band intact.

Topological properties of nearly flat bands in bilayer $\alpha-\mathcal{T}3$ lattice