We report high-precision measurements on the thallium fluoride (J) over tilde = 1 hyperfine manifold of the B-3 Pi(1) (v = 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 (J) over tilde = 1 manifold of hyperfine level splittings. We observe Stark shifts of the (J) over tilde = 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) and 17.4(11) MHz, respectively.