Kurosawa's Yojimbo was inspired in part by the work of Dashiell Hammett. It had a seemingly amoral hero who deals with criminals on their own terms and defeats them. Part of the plot was inspired directly by The Glass Key. And it was remade as A Fistful of Dollars, so the private eye genre did have an indirect influence on westerns in the 1960s.
And, before that, there was Shotgun Slade, a half-hour TV western that aired from 1959-1961.
Apparently private eye shows were edging out westerns on TV at the time, so they compromised. They made a western about a "private detective" in the old west whose weapon is a shotgun-rifle combination.
Had a jazz soundtrack which was pretty good for a western. It really wasn't bad. In an episode I saw, Shotgun Slade brings a weaselly guy who embezzled thousands of dollars from a children's charity to justice and thwarts some train robbers in the process.
Might have been just as well without the combination shotgun/rifle gimmick, though. For one thing, he has to walk around carrying this thing all the time.
Again, the show is public domain. Available on streaming video.