The blazar Mrk 421 shows frequent, short flares in the TeV energy regime. Due to the fast nature of such episodes, we often fail to obtain sufficient simultaneous information about flux variations in several energy bands. To overcome this lack of multi-wavelength (MWL) coverage, especially for the pre- and post-flare periods, we have set up a monitoring program with the FACT telescope (TeV energies) and the Neil Gehrels Swift Observatory (X-rays). On 2019 June 9, Mrk 421 showed a TeV outburst reaching a flux level of more than two times the flux of the Crab Nebula at TeV energies. We acquired simultaneous data in the X-rays with additional observations by XMM-Newton and INTEGRAL. For the first time, we can study a TeV blazar in outburst taking advantage of highly sensitive X-ray data from XMM-Newton and INTEGRAL combined. Our data set is complemented by pointed radio observations by Effelsberg at GHz frequencies. We present our first results, including the gamma-ray and X-ray light curves, a timing analysis of the X-ray data obtained with XMM-Newton, as well as the radio spectra before, during and after the flare.