  We study the effect of Haldane flux in the bilayer $\alpha$-$\mathcal{T}_3$
lattice system, considering possible non-equivalent, commensurate stacking
configurations with a tight-binding formalism. The bilayer
$\alpha$-$\mathcal{T}_3$ lattice comprises six sublattices in a unit cell, and
its spectrum consists of six bands. In the absence of Haldane flux, threefold
band crossings occur at the two Dirac points for both valence and conduction
bands. The introduction of Haldane flux in a cyclically stacked bilayer
$\alpha$-$\mathcal{T}_3$ lattice system separates all six bands, including two
low-energy, corrugated nearly flat bands, and assigns non-zero Chern numbers to
each band, rendering the system topological. We demonstrate that the
topological evolution can be induced by modifying the hopping strength between
sublattices with the scaling parameter $\alpha$ in each layer. In the dice
lattice limit ($\alpha = 1$) of the Chern-insulating phase, the Chern numbers
of 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 between
the two lower energy bands. These crossings occur at different values for the
conduction and valence bands and depend further on the next nearest neighbor
(NNN) hopping strength. At the transition point, the Chern numbers of the two
lower conduction and valence bands change discontinuously from $\pm 2$ to $\pm
5$ and $\pm 3$ to $0$, respectively, while leaving the Chern number of the
third band intact.
