We present observations of a blowout jet that experienced two distinct ejection stages. The first stage started from the emergence of a small positive magnetic polarity, which was cancelled by the nearby negative magnetic field and caused the rising of a mini-filament and its confining loops. This further resulted in a small jet due to the magnetic reconnection between the rising confining loops and the overlying open field. The second ejection stage was mainly due to successive removal of the confining field by reconnection: the filament erupted, and the erupting cool filament material directly combined with the hot jet that originated form the reconnection region and therefore formed the cool and hot components of the blowout jet. During the two ejection stages, cool Ha jets are also observed cospatial with their coronal counterparts, but their appearance times are earlier by a few minutes than those of the hot coronal jets. The hot coronal jets are therefor possibly caused by the heating of the cool Ha jets or the rising of the reconnection height from the chromosphere to the corona. The scenario that magnetic reconnection occurred between the confining loops and the overlying open loops is supported by many observational facts, including the bright patches on both sides of the mini-filament, hot plasma blobs along the jet body, and periodic metric radio type III bursts at the very beginnings of the two stages. The evolution and characteristics of these features show the detailed nonlinear process in magnetic reconnection.