Kill Me Tomorrow — Richard S. Prather

In the retirement wastelands of Arizona, Shell Scott does a favor for an Italian starlet
 
The bar goes silent when actress Lucrezia Brizante enters. Shell Scott has no idea why this Italian beauty is walking toward him, but he's about to find out. As it happens, he's exactly what she's looking for. There are thousands of private detectives in the United States, but none as batty as Scott, and for what Lucrezia desires, battiness is required.
 
Lucrezia is about to start filming Sins of Caesar's Orgies, the most outrageous movie of her career, but all she can think about is her father, who has joined the horde of retirees flocking to the deserts of Arizona. Something has soured his golden years, and he can do nothing but mutter to himself about some vague, evil conspiracy. Scott doesn't really care what's going on. All he knows is that Lucrezia has eyes as dark as tar pits, and he's ready to hurl himself inside them.

Kill Me Tomorrow is the 35th book in the Shell Scott Mysteries, but you may enjoy reading the series in any order.