Terahertz-based electron acceleration has recently emerged as a promising candidate for driving next-generation highbrightness electron sources. Although initial demonstrations have proven the feasibility of this technology for accelerating and manipulating the phase space of electrons, further demonstrations of exquisite timing control are required to make use of terahertz acceleration for demanding applications such as light sources and ultrafast electron diffraction. In this paper, we use a two-stage segmented-terahertz-electron-accelerator-and-manipulator (STEAM) setup to demonstrate control over the electron beam energy, energy spread, and emittance. The first rebunching stage is used to tune the duration of 55 keV electron bunches from a DC electron gun that enables femtosecond phase control at the second accelerating stage. For optimized parameters, energy spread and emittance are reduced by 4× and 6×, respectively, relative to operation with the first stage off. A record energy gain of ∼70 keV was achieved at a peak accelerating field of 200 MV/m, resulting in a >100% energy boost in a terahertz-powered accelerator for the first time. These results represent a critical step forward for the practical implementation of terahertz-powered devices in ultrafast electron sources.