G for Gumshoe by Sue Grafton

Three things happen to Kinsey Millhone on her thirty-third birthday: she moves into her remodeled apartment, which has finally been finished; she is hired by Irene Gersh, a sickly Santa Teresa resident, to head out to the Slabs in the Mojave Desert and locate her mother; and she gets the news that Tyrone Patty, a particularly dangerous criminal she helped the Carson City, Police Department track down a few years back, has hired a hit-man to kill her.