  We report on the improved measurement of the hyperfine structure of
$4f^{12}(^3 H_6)5d_{5/2}6s^2$ $(J=9/2)$ excited state in Tm-169 which is
involved in the second-stage laser cooling of Tm. To measure the absolute value
of the hyperfine splitting interval we used Doppler-free frequency modulation
saturated absorption spectroscopy of Tm atoms in a vapor cell. The sign of the
hyperfine constant was determined independently by spectroscopy of laser cooled
Tm atoms. The hyperfine constant of the level equals $A_J=-422.112(32)$ MHz
that corresponds to the energy difference between two hyperfine sublevels of
$-2110.56(16)$~MHz. In relation to the saturated absorption measurement we
quantitatively treat contributions of various mechanisms into the line
broadening and shift. We consider power broadening in the case when Zeeman
sublevels of atomic levels are taken into account. We also discuss the line
broadening due to frequency modulation and relative intensities of transitions
in saturated-absorption experiments.
