We investigated generation of pre-formed plasma on plates and inside cone targets due to a pre-pulse before the arrival of the main ultra-intense laser pulse in the fast-ignition scheme of the inertial confinement fusion. We estimated the pre-pulse level to be 130 mJ for LFEX laser used in the 2009 FIREX experiment, and the density gradient scale length of the pre-formed plasma inside the cone target to be 27–47 microns between the critical and 1/10 of the critical density, based on the two-dimensional radiation hydrodynamic simulations. In order to reduce the generation of pre-formed plasma, we investigated a thin CH foil pre-pulse absorber, and proposed using a cone target with a pointed tip. We simulated CH plasma expansion to show that the CH foils works as a pre-pulse absorber. We also show the aluminum pointed tip of the cone target can delay the shock arrival time by 20 ps, much longer than the delay for the 10 micron thickness gold tip used in the typical implosion of GXII at Osaka University.