Bright single photon sources have recently been obtained by inserting solid-state emitters in microcavities. Accelerating the spontaneous emission via the Purcell effect allows both high brightness and increased operation frequency. However, achieving Purcell enhancement is technologically demanding because the emitter resonance must match the cavity resonance. Here, we show that this spectral matching requirement is strongly lifted by the phononic environment of the emitter. We study a single InGaAs quantum dot coupled to a micropillar cavity. The phonon assisted emission, which hardly represents a few percent of the dot emission at a given frequency in the absence of cavity, can become the main emission channel by use of the Purcell effect. A phonon-tuned single photon source with a brightness greater than 50% is demonstrated over a detuning range covering 10 cavity line widths (0.8 nm). The same concepts applied to defects in diamonds pave the way toward ultrabright single photon sources operating at room temperature.