  We report high-precision measurements on the thallium fluoride $\tilde{J} =
1$ hyperfine manifold of the $B^3\Pi_1$ ($\nu = 0$) state. This state is of
special interest because it is central to an optical cycling scheme that is
envisioned to play an important role in enhancing the sensitivity of the
CeNTREX nuclear Schiff-moment experiment presently under construction. The
measurements are made by monitoring the fluorescence induced by narrow-band
laser excitation of a cryogenic molecular beam. We use a multipass arrangement
of the laser beam to enhance fluorescence. When viewed with a camera, we can
spatially resolve images from adjacent passes that approach the molecules from
opposing directions. These images yield a sensitive visual method to identify
the central frequency of a transition. Coupling these line-center
determinations with frequency calibration from an acousto-optic modulator has
allowed a more precise determination of the $\tilde{J} = 1$ manifold of
hyperfine level splittings. We observe Stark shifts of the $\tilde{J} = 1$
levels and infer a permanent electric dipole moment of 2.28(7) D and
$\Lambda$-doublet splittings for the $F_1' = 1/2$ and $F_1' = 3/2$ manifolds of
14.4(9) MHz and 17.4(11) MHz, respectively.
