  Following a recent paper by Fodor et al. (arXiv:1406.0827), we reexamine
several types of tree-level improvements on the flow action with various gauge
actions in order to reduce the lattice discretization errors in the Yang-Mills
gradient flow method. We propose two types of tree-level, $\mathcal{O}(a^4)$
improved lattice gradient flow including the rectangle term in both the flow
and gauge action within the minimal way. We then perform numerical simulations
with the simple plaquette gauge action for testing our proposal. Our numerical
results of the expectation value of the action density, $\langle E(t)\rangle$,
show that two $\mathcal{O}(a^4)$ improved flows significantly eliminate the
discretization corrections in the small flow time $t$ regime. On the other
hand, the values of $t^2\langle E(t)\rangle$ in the large $t$ regime, where the
lattice spacing dependence of the tree-level term dies out as inverse powers of
$t/a^2$, are different between the results given by two optimal flows leading
to the same $\mathcal{O}(a^4)$ improvement at tree level. This may suggest that
non-negligible $\mathcal{O}(g^2 a^2)$ effect sets in the large $t$ regime,
where the running coupling $g(1/\sqrt{8t})$ becomes large.
