Methods and systems are provided for operating a fuel system in an engine, the fuel system including a supply pump for delivering fuel to the fuel system and pressurizing fuel received from a feed pump, a fuel tank, a fuel filter for filtering fuel, a fuel rail, and a fuel injector. One example method comprises, during an engine cold-start, operating the supply pump, and adjusting a supply pump operation mode between at least a pressure-controlled mode and a volume-controlled mode based on a fuel temperature and pressure.