Precision spectroscopy of trapped HfF ⁺ will be used to search for the permanent electric dipole moment of the electron (eEDM). Prior to this study, spectroscopic information necessary for state preparation, readout, and analysis of systematic errors was not available. We have developed a powerful technique for broadband, high-resolution survey spectroscopy of molecular ions that combines cavity-enhanced direct frequency-comb spectroscopy with velocity-modulation spectroscopy (vms) and used this to measure four bands in HfF ⁺ over a 1000 cm ^-1 bandwidth near 800 nm. Additionally, we performed targeted scans with cw-laser vms to find 15 additional bands from 9950 to 14600 cm ^-1. We present a detailed analysis of these bands to obtain high-precision rovibrational constants, Λ-doublings, and isotope splittings for eight electronic states. We also use our results to improve theoretical predictions and discuss implications of our measurements to the eEDM experiments. These results demonstrate the application of frequency-comb and cw-vms for broadband, high-resolution spectroscopy of molecular ions.