  We investigate the finite-size-scaling (FSS) behavior of the leading Fisher
zero of the partition function in the complex temperature plane in the
$p$-state clock models of $p=5$ and $6$. We derive the logarithmic finite-size
corrections to the scaling of the leading zeros which we numerically verify by
performing the higher-order tensor renormalization group (HOTRG) calculations
in the square lattices of a size up to $128 \times 128$ sites. The necessity of
the deterministic HOTRG method in the clock models is noted by the extreme
vulnerability of the numerical leading zero identification against stochastic
noises that are hard to be avoided in the Monte-Carlo approaches. We
characterize the system-size dependence of the numerical vulnerability of the
zero identification by the type of phase transition, suggesting that the two
transitions in the clock models are not of an ordinary first- or second-order
type. In the direct FSS analysis of the leading zeros in the clock models, we
find that their FSS behaviors show excellent agreement with our predictions of
the logarithmic corrections to the Berezinskii-Kosterlitz-Thouless ansatz at
both of the high- and low-temperature transitions.
