Spreading time, the time that an impacting droplet attains the maximum wetting area on a solid surface, plays a critical role in many engineering applications particularly where heat transfer or chemical reactions are involved. Although the impact dynamics of a droplet significantly differ across the different spreading regimes depending on various collision parameters, it still remains unclear (i) how the spreading time changes for each spreading regime, and (ii) how the target curvature can affect the spreading time. In the present study, the spreading time during droplet impact on a spherical target is systematically studied at the three different spreading regimes for a wide range of impact parameters (Weber number, equilibrium contact angle, Ohnesorge number and droplet-to-target size ratio). The changes of spreading time depending on the impact parameters and underlying physical mechanisms are analyzed in detail at the level of three different spreading regimes. Our results show that the spreading time, proper time scales, dominant impact parameters and associated physical behaviors all significantly and non-linearly change across the three spreading regimes. An improved prediction model for the spreading time is also proposed for each regime, which is now based on only the controllable variables and has an explicit form. The effect of target curvature on the spreading time is further investigated, and finally, a data-driven prediction model is proposed to represent the complicated and non-linear nature of the spreading time broadly across the three spreading regimes.