For the identification of low momentum charged particles and for event timing purposes a barrel Time-of-Flight (TOF) detector surrounding the interaction point is planned for the PANDA experiment at FAIR. Since the boundary conditions in terms of available radial space and radiation length are quite strict the favored layout is a hodoscope composed of several thousand small scintillating tiles (SciTils) read out by silicon photomultipliers (SiPMs). A time resolution of well below 100 ps is aimed for. With the originally proposed 30 x 30 x 5 mm3 SciTils read out by two single 3 x 3 mm2 SiPMs at the rims of the scintillator the targeted time resolution can be just reached, but with a considerable position dependence across the scintillator surface. In this paper we discuss other design options to further improve the time resolution and its homogeneity. It will be shown that wide scintillating rods (SciRods) with a size of, e.g., 50 x 30 x 5 mm3 or longer and read out at opposite sides by a chain of four serially connected SiPMs a time resolution down to 50 ps can be reached without problems. In addition, the position dependence of the time resolution is negligible. These SciRods were tested in the laboratory with electrons of a 90Sr source and under real experimental conditions in a particle beam at CERN. The measured time resolutions using fast BC418 or BC420 plastic scintillators wrapped in aluminum foil were consistently between 45 and 75 ps dependent on the SciRod design. This is a significant improvement compared to the original SciTil layout.