Big plot hole. Beware -- big spolier as to how case solved.
***BIG SPOILER TO ENTIRE MOVIE*********
With all the big talent that went into making this movie, I am surprised that they came up with such a huge PLOT HOLE in solving the crime. I hope that I am wrong somehow; if so, let me know via a "reply" here.
BACKGROUND:
(1) The Anne Baxter character went on a first date with Burr, got drunk, went to his apartment, fought off his advances, struck him with a poker, passes out. and then wakes up and leaves without looking about to see if Burr is there, dead or alive. Later she finds out that Burr was killed in the apartment with the poker, is afraid she had killed him, but doesn't remember.
(2) It happens that after Baxter left Burr's apartment, the real murderer (a jilted, pregnant woman) came by and killed Burr with the same poker because he wouldn't marry her.
(3) Burr was in the habit of playing musical records to his women friends. He liked playing the music of their first date. Baxter told Conte (who was helping her) that Burr played "The Blue Gardenia" music on her visit to his apartment, which was playing when she struck him with the poker. When Burr's pregnant "girlfriend" came by later that night, he tried to soothe her by playing a different music, the music he played on their first date.
(4) Late in the movie, Conte realizes that when Burr's body was found, "Blue Gardenia" was not in the record player; rather it was the other music for the pregnant "girlfriend." While this wouldn't help her in court, it was enough to convince Conte and Baxter that she almost certainly didn't murder Burr, that in any event someone (almost definitely Burr) changed the record after Baxter left the apartment that night. OK, good so far; I found that part neat.
PLOT HOLE:
The next action, which led to the discovery of the real murderer, is what I find very questionable and call a plot hole. Conte and the cops look at the record album cover of the second music (that Burr played for the pregnant "girlfriend") and see where Burr bought it -- the cover was stamped with the local store's name and address. There they discover and scare (just by their presence) the female store clerk (who turned out to be --hold onto your hats-- the pregnant "girlfriend"!) who sold the record to Burr and who killed him that night! Hard as I try, I can't imagine any investigator going to the record store in that situation. It would make just as much sense to go to the store where Burr bought the record player.
Again, if you disagree, let me know with reply to my post.