We report here an experimental setup to perform three-pulse pump-probe measurements over a wide wavelength and temperature range. By combining two pump pulses in the visible (650-900 nm) and mid-IR (5-20 $mu$m) range, with a broadband supercontinuum white-light probe, our apparatus enables both the combined selective excitation of different material degrees of freedom and a full time-dependent reconstruction of the non-equilibrium dielectric function of the sample. We describe here the optical setup, the cryogenic sample environment and the custom-made acquisition electronics capable of referenced single-pulse detection of broadband spectra at the maximum repetition rate of 50 kHz, achieving a sensitivity of the order of 10$^{-4}$ over an integration time of 1 s. We demonstrate the performance of the setup by reporting data on mid-IR pump, optical push and broadband probe in a single-crystal of Bi$_2$Sr$_2$Y$_{0.08}$Ca$_{0.92}$Cu$_2$O$_{8+delta}$ across the superconducting and pseudogap phases.
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