The high-order mode in a fishbone waveguide accommodates the easy manipulation of group index dispersion. To minimize Fresnel reflection, group velocity mismatch can be compensated for by a casual step taper, but with low coupling efficiency. Here, we propose a mode converter to boost the coupling efficiency to in and out of the slow-light fishbone waveguide. Split, phase, and side slots in a mode converter are employed to compensate for the mode mismatch in terms of the width of the lobes, the phase offset, and the power ratio between neighboring lobes. A high coupling efficiency of >70% leads to the efficient compact true-time delay required for phased array antennas.