  In this paper, we investigate the second law of the black holes in Lovelock
gravity sourced by a conformally coupled scalar field under the first-order
approximation when the perturbation matter fields satisfy the null energy
condition. First of all, we show that the Wald entropy of this theory does not
obey the linearized second law for the scalar-hairy Lovelock gravity which
contains the higher curvature terms even if we replace the gravitational part
of Wald entropy with Jacobson-Myers (JM) entropy. This implies that we cannot
naively add the scalar field term of the Wald entropy to the JM entropy of the
purely Lovelock gravity to get a valid linearized second law. By rescaling the
metric, the action of the scalar field can be written as a purely Lovelock
action with another metric. Using this property, by analogy with the JM entropy
of the purely Lovelock gravity, we introduce a new formula of the entropy in
the scalar-hairy Lovelock gravity. Then, we show that this new JM entropy
increases along the event horizon for Vaidya-like black hole solutions and
therefore it obeys a linearized second law. Moreover, we show that different
from the entropy in $F($Riemann$)$ gravity, the difference between the JM
entropy and Wald entropy also contains some additional corrections from the
scalar field.
