The supersonic base flow with an exhaust jet is significantly unsteady, whose characteristics can be influenced by the exhaust jet states. By the exhaust jet, we mean the working fluid produced by the rocket engine and exhausted by the base nozzle in supersonic velocity, which is used to generate thrust. A numerical investigation of the base flow at Mach 1.96 is carried out via the delayed detached eddy simulation. The effects of presence, expanded states, and temperature of the exhaust jet on the unsteady characteristics are analyzed. Compared with the exhaust jet off, the exhaust jet presence causes the base pressure fluctuation intensity to increase in the high-frequency band (SrD > 0.5) and changes the dominant mode from wake oscillation (SrD = 0.12) to the periodically expand-shrink of the exhaust jet diameter (SrD = 0.16). Compared with the underexpanded state, the overexpanded exhaust jet induces the flow separation inside the nozzle, which causes a decrease in the base pressure fluctuation intensity in the high-frequency band (SrD > 1.0), and the separation point movement and exhaust jet oscillation constitute the dominant mode (SrD = 0.13). Compared with the low-temperature jet, the base pressure fluctuation intensity is slightly reduced in the high-frequency band (SrD > 1.0) for the high-temperature jet, and the flapping motion of the jet shear layer becomes the dominant mode (SrD = 1.5).