  We analyze second-order $M1$-$M1$ and $M1$-$E2$ effects to the hyperfine
structure (HFS) of the lowest energy $P$ states of alkali-metal atoms arising
from the coupling of the two ($J=1/2,3/2$) fine-structure levels through the
hyperfine interaction. We find these effects to be especially sizable in Li,
leading to a $9\sigma$ correction to the most accurate reported experimental
value of the $A(P_{1/2})$ HFS constant of $^7$Li [D. Das and V. Natarajan, J.
Phys. B \textbf{41}, 035001 (2008)]. For the remaining alkali-metal systems,
the results tabulated within may be referenced as higher precision is sought in
experimental determination of the HFS constants.
