This part of Poe's story seems weak. If Minister D-- is so terribly clever, he certainly ought to be suspicious when Dupin visits him, not once but twice, for no apparent specific reason but just ostensibly to have a casual conversation. The Minister ought to have some notion that Dupin has had dealings with the Parisian police and even that he has assisted the Prefect of Police. On his first visit to the Minister which is supposed to be "quite by accident," Dupin is wearing "a pair of green spectacles." Surely that would make Dupin look weird and give the Minister cause to wonder what his visitor is up to, especially since the blackmailer is well aware that he is under heavy surveillance and that his rooms have been meticulously searched at least twice. It was odd enough for Dupin to pay the first visit unannounced and for no apparent purpose, but when he came back the second time that should have been sufficient to make the Minister more than merely suspicious but positive that Dupin was after the purloined letter.
At the conclusion of the story Dupin tells his friend:
D--, at Vienna once, did me an evil turn, which I told him, quite good-humoredly, that I should remember.
The Minister should remember (1) that he did Dupin an evil turn, and (2) that Dupin promised that he would get back at him someday. These are additional reasons why the Minister should be on guard.
It is hard enough to believe that the police would not find a letter hidden in plain sight, but Dupin's two visits to the Minister make the story even harder to believe. "The Purloined Letter" does not measure up to Poe's "The Gold-Bug" or "The Murders in the Rue Morgue," both of which are more logical and more dramatic.