Laser speckle contrast imaging is a technique to determine blood flow rate with a limitation of low dynamic range. In this Letter, we introduce a varied illumination speckle contrast imaging method. It utilizes varying illumination during exposure to customize the correlation time (flow rate) to speckle contrast relation. The method can cover an order of magnitude larger range flow rate in a single exposure compared to constant illumination methods. The proposed method enables high dynamic range flow rate imaging, which is advantageous in studying larger vessels and small arteries. We demonstrate the theory by simulations and ex vivo and in vivo measurements.