  We investigate topological characteristics of the photon-dressed band
structure of $\alpha$-$\mathcal{T}_3$ lattice on being driven by off-resonant
circularly polarized radiation. We obtain exact analytical expressions of the
quasienergy bands over the first Brillouin zone. The broken time-reversal
symmetry caused by the circularly polarized light lifts the triple point
degeneracy completely at both the Dirac points. The gaps become unequal at $
{\bf K}$ and ${\bf K}^{\prime}$ (except at $\alpha=0$ and 1), which reveals the
absence of inversion symmetry in the system. At $\alpha=1/\sqrt{2}$, the gap
between flat and valence bands closes at ${\bf K}$, while that between
conduction and flat bands closes at ${\bf K}^{\prime}$, thereby restoring a
semimetalic phase. At the gap closing point ($\alpha=1/\sqrt{2}$) which is
independent of the radiation amplitude, there is a reappearance of low-energy
Dirac cones around ${\bf K}$ and ${\bf K}^{\prime}$ points. Under the influence
of the circularly polarized radiation, the $\alpha$-$\mathcal{T}_3$ lattice is
transformed from semimetal to a Haldane-like Chern insulator characterized by
non-zero Chern number. The system undergoes a topological phase transition from
$\mathcal{C} = 1 (-1)$ to $\mathcal{C}=2 (-2)$ at $\alpha =1/\sqrt{2}$, where
$\mathcal{C}$ is the Chern number of the valence (conduction) band. This sets
an example of a multiband system having larger Chern number. These results are
supported by the appearance of chiral edge states in irradiated
$\alpha$-$\mathcal{T}_3$ nanoribbon.
