The hyperfine coupling constants of the proton dripline odd-odd $^{40}\mathrm{Sc}$ nucleus were deduced from the hyperfine spectrum of the $3d4s{\phantom{\rule{0.16em}{0ex}}}^{3}{\text{D}}_{2}\ensuremath{\leftrightarrow}3d4p{\phantom{\rule{0.16em}{0ex}}}^{3}{\text{F}}_{3}^{\ensuremath{\circ}}$ transition in Sc II, measured by the bunched beam collinear laser spectroscopy technique. The ground-state magnetic-dipole and electric-quadrupole moments were determined as $\ensuremath{\mu}=+5.57(4)(2){\ensuremath{\mu}}_{N}$ and $Q=+42(38)(28)\phantom{\rule{4pt}{0ex}}{e}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{fm}}^{2}$, respectively. The magnetic moment is well reproduced by the additivity rule with magnetic moments of neighboring odd-even nuclei in the vicinity of the doubly magic $^{40}\mathrm{Ca}$ nucleus. An ab initio multishell valence-space Hamiltonian was also employed to calculate the magnetic moment of $^{40}\mathrm{Sc}$, which spans across the $sd$ and $fp$ nuclear shells, where we obtained good agreements.