The Lonely Profession(1969)

Private investigator Leo Gordon is hired to trail Karen Mendaros, the mistress of a reclusive billionaire. When they meet, Gordon and Mendaros hit it off and check in at a motel. Gordon wakes up the next morning and discovers that Mendaros had been murdered during the night. Gordon opens his own investigation of Mendaros' past in an attempt to determine who killed Mendaros and why he's been set up as the fall guy.