  The refined values of the magnetic octupole moments of $^{113}$In and
$^{115}$In are obtained by combining high-precision atomic calculations with
corresponding hyperfine structure spectrum. We performed an \textit{ab initio
}calculations of hyperfine-structure properties for the low-lying states of In
atom using the single and double approximated relativistic coupled-cluster
method. The hyperfine-structure properties includes first-order
hyperfine-structure constants and the second-order magnetic dipole-magnetic
dipole, magnetic dipole-electric quadrupole, and electric quadrupole-electric
quadrupole effects caused by the off-diagonal hyperfine interaction. Based on
our theoretical results, we reanalyze the previously measurements of hyperfine
splitting in the 5$p_{3/2}$ state of $^{113}$In and $^{115}$In [Eck and Kusch,
Phys. Rev. 106, 958 (1957)], determining corresponding hyperfine-structure
constants $A$, $B$, and $C$. By combining these undated HFS constants and our
theoretical results, the magnetic octupole moments of $^{113}$In and $^{115}$In
nuclei are extracted to be $\Omega(^{113}\rm
In)=0.455(44)$~$\mathrm{\mu_{N}\times b}$ , and $\Omega(^{115}\rm
In)=0.443(42)$~$\mathrm{\mu_{N}\times b}$, respectively. The refined values of
magnetic octupole moments are about smaller 21\% than the previously reported
results by Eck and Kusch [Phys. Rev. 106, 958 (1957)]. Additionally, we also
determine the electric quadrupole moment of $^{115}$In nuclei to be
$Q(^{115}\rm In)=0.767(9)$ b by combining our theoretical results and the
measured values for hyperfine-structure constants of the 5$p_{3/2}$ and
6$p_{3/2}$ states. Our results are compared with available experimental and
theoretical results.
