I'm going to leave the question of whether late 19th Century or early 20th Century police forces ever turned to real-life individuals like Sherlock Holmes to another expert, but one bit of historical insight I can put into play here is that it is critically important to understand a major issue that complicates the question a great deal, namely, that police as we understand them in the early 21st Century were a relatively new part of everyday life in the late 19th Century in Europe and the United States.

What was even newer still was the idea of doing systematic criminal investigation using forensic techniques or gathering evidence methodically. (Arthur Conan Doyle has Sherlock Holmes saying as much frequently in the stories.) As the historian Eric Monkkonen pointed out in his book Police in Urban America, 1860-1920, "For us to comprehend the blundering, ignorance, inefficiency, incompetence, and general confusion of the nineteenth-century police when dealing with crime requires an imaginative leap over a great distance." In the new mid-19th Century police forces established in Western nations, "the art of criminal detection bore more than a little resemblance to divination" and the new police "spent the most useful of their long hours on duty reporting open sewers, shooting stray dogs, and arresting drunks".

Monkkonen is laying it on pretty thick, but he's describing a general consensus among historians about the origins of modern police forces. Nobody had a "proper police background" as we might think of it today until later in the 20th Century, including even the specific legal rules or permissions to operate as authorized agents of municipal or local governments. (As Monkkonen puts it, 19th Century police were "semiliterate members of the working class wearing outfits that they thought looked like servants' livery, charged with duties which no one clearly understood".)

So in some sense the question the OP is asking can really only be posed in a more systematic way about the circumstances under which mid to late 20th Century police or law enforcement have brought in outside consultants into investigations, and when those consultants might have been licensed as private detectives (or other professionals with some connection to the legal system like bounty hunters etc.)