Terahertz (THz) time-domain spectroscopy (TDS) is a sensitive approach to material characterization. It critically relies on a sufficiently large bandwidth, which is not straightforwardly available in typical THz-TDS systems that are often limited to below 3 THz. Here, we introduce a hybrid THz-source concept based on a spintronic THz emitter (STE) deposited onto a thin, free-standing GaSe nonlinear crystal. By tuning the magnetic state and the phase-matching parameters of the hybrid emitter, we generate an ultrabroadband spectrum covering the full range from 1 to 40 THz. We achieve significantly enhanced spectral amplitudes above 10 THz compared to the bare STE, resulting in ultrashort THz-pulse durations down to 32 fs. Finally, we demonstrate the straightforward tunability of the shape of the few-cycle pulse from symmetric to antisymmetric.