  Optical lattice clocks are the prospective devices that can probe many subtle
physics including temporal variation of the fine structure constant
($\alpha_e$). These studies necessitate high-precision measurements of atomic
clock frequency ratios to unprecedented accuracy. In contrast to the earlier
claimed highest sensitive coefficient ($K$) clock transition to $\alpha_e$ in
Yb [Phys. Rev. Lett. $\textbf{120}$, 173001 (2018)], we found the
$4f^{14}6s6p\,(^3P_2) - 4f^{13}5d6s^{2}\,(^3P_2^*)$ transition of this atom can
serve as the clock transition with the largest $K$ value (about $-$25(2)).
Moreover, we demonstrate a scheme to attain simultaneous magic trapping
conditions for this clock transition with the other two proposed clock
transitions $4f^{14}6s^2\,(^1S_0) - 4f^{14}6s6p\,(^3P_2)$ and
$4f^{14}6s^2\,(^1S_0) - 4f^{13}5d6s^{2}\,(^3P_2^*)$; also exhibiting large $K$
values. This magic condition can be realized by subjecting Yb atoms to a bias
magnetic field at a particular polarization angle along the quantization axis
in an experimental set up. Upon realization, it will serve as the most
potential optical lattice clock to probe $\alpha_e$ variation.
