Pedestrican struck by a vehicle going on reverse. If we ask for compensation with the car insurance pay for it or the driver personally?

My mother was crossing the street ( one way) when she got hit in her face and arm by an SUV that was driving on reverse. She suffered broken nose, arm and toes plus swollen in her feet. The car insurance, Geico called and said that they would cover medical bills up to $50 000. If my mother needs surgery, we expect the medical bills to be higher. If we hire a lawyer we would like compensation only from Geico. We do not want to get a compensation directly from the driver. Is this possible? Also, the lawyer we met with said that he would be getting 1/3 of the compensation. Is this a set rate? Any info, we would appreciate. Thank you.