murray johnc,
Yes, absolutely. Reverse directories such as the "Haynes Criss+Cross (tm)" and similar directories from other publishers have been around for many decades in the US, going back at least to the 1940's to my knowledge. These directories were available on a subscription basis, updated quarterly, and were used by debt collection agencies, direct marketers, police departments, and private investigators (like me). Many public libraries also had reverse directories available in the reference section. You could look up listings by name (last name first, of course), address (which would show all listed telephone numbers/names at a given address [useful for apartments]), and by telephone number. Detectives often referred to the reverse directories as "blue books" because of the blue pages on which they were printed, and even used "blue-book" as a verb, as in, "hey dispatch, can you blue-book a number for me"? These directories were published on a citywide basis, just like conventional phone books; but as a subscriber, you could call the publisher during normal business hours, and have them look up a number or address for you in a different city. Another interesting aspect of reverse directories is that because they obtained their data from the phone companies, non-published and unlisted numbers did not appear.
A similar product available during those halcyon days long before the Internet was the "city directory", which DID contain most non-published and unlisted telephone numbers, along with a virtual goldmine of information, including names all adult members (and sometimes even children) in the household, plus the occupation and employer of employed adults in the household. How did they get all this sensitive information? They asked for it! City Directory publishers sent armies of canvassers door to door, who either charmed or cajoled most people into willingly giving up all this very personal information, even those people who had actually paid an extra fee to the telephone company to have their number non-published or unlisted!
As to your main question, it's been a long time since I've seen "Klute", but I would speculate that after he got the forwarding number, he called a little known number available to phone company technicians (and wily investigators/detectives) called a "CNA" line. If you had the number, you would simply call it up, impersonate a telco employee, and ask the operator at the other end for a CNA (Customer Name and Address) for the number in question, e.g., "Hi, this is Jim Bob at frames. I need a CNA on 212-555-XXXX". You might occassionally be asked for an employee number (always a 5 digit number back in those days), and you could usually make one up and still get the information. The trick is that the operator didn't actually have a way of verifying your bona fides, she (always "she" back then) wanted to see if you would come back with it immediately or choke on your reply. The great thing about CNA lines is that they were always up to date.
In another classic movie of the era, "Three Days of the Condor", Robert Redford's character played the dialed number portion of a tape from a wiretap back to a semi-automated system to first decode the DTMF (touch tones) digits dialed, and then spoke to an operator to obtain the CNA. This system was operated not by the telephone company, but by "THE Company", the CIA.
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