"I think I know a lost cause when I see one."
-- Albus Dumbledore, standing up to leave Slughorn's home (HBP4)
July 13, 1996, just after midnight – En route to the Burrow from Privet Drive, Dumbledore takes Harry to call on Horace Slughorn – who, in trying to evade the Death Eaters, possible employers, and anyone who would make him do work or associate himself with one side over another in a conflict, has been moving to a different Muggle home every few days. Knowing that Dumbledore is coming to ask him to return to teaching at Hogwarts, Slughorn quickly stages his temporary residence so that it appears as though a violent intrusion by Death Eaters has occurred. However, Slughorn forgets to put a Dark Mark in the sky above the house, and it is apparent to Dumbledore that it is a ruse.
Inside, Dumbledore easily locates the armchair that is really Slughorn in disguise, and turns him back to his human shape. After the two old colleagues clean up the mess with magic and Slughorn steadfastly refuses to go back to teaching, Dumbledore leaves Slughorn and Harry alone. Slughorn boasts to Harry about successful former students of his with whom he keeps in touch and gets perks. He also displays a questionable attitude regarding Muggle-borns; apparently liking and accepting individual ones (such as Harry’s mother), but, on the whole, doubting their competency as witches and wizards. Just as Harry and Dumbledore are about to leave when Slughorn decides that “collecting” Harry as one of his favorites and being safe at Hogwarts outweigh the downsides of Dumbledore’s request, and he takes the job. But he says loudly as Harry and Dumbledore are leaving, “I’ll want a pay raise!”
Dumbledore called on the Dursleys at 11 pm. After that conversation, he and Harry left for Budleigh Babberton. Upon arrival, Harry noticed that "(a)ccording to a clock on a nearby church, it was almost midnight." This means that they arrive at the house where Slughorn is staying around midnight and visit between 12 and 1 am on July 13.
Note that this date does not try to reconcile with the fact that according to Pottermore, Sirius Black was killed on Thursday, June 18, which in reality was a Tuesday. This would change the date of the Saturday of visiting Slughorn to July 11. Either date could be considered correct, assuming that Pottermore's date for Sirius's death is in fact canon (which is not certain). Thanks to user ironyak1 for pointing out this discrepancy.
As Dumbledore and Harry walk to the house in which Slughorn is staying, Dumbledore remarks that Voldemort "killed enough people to make an army of." In the sea cave months later, they discover that Voldemort does, indeed, have quite a sizable group of corpses ready to do his bidding (HBP26). -BB