The aftermath of the Dillinger shooting revealed, as expected, the usual confusion which comes immediately after a law enforcement shooting. On July 22, after the shooting, Hoover spoke to FBI Inspector, Samuel Cowley in Chicago and it's clear that the immediate belief was that Winstead may have been the only agent who fired at Dillinger. We know this was clarified in the ensuing days to include SA's Hurt and Hollis also. Regardless, Hoover's call to Cowley is here regarding "credit" for the police and keeping Winstead away from the press.
On July 23, 1934 J. Edgar Hoover made it very clear to his Assistant Directors, and the team at Chicago, FBI what his thoughts were on the coming media coverage of the Dillinger shooting, however he would comply with any wishes of the Attorney General:
It's now known that on the same day, Asst. Attorney General William Stanley came up with an idea for Attorney General Homer Cummings who was crossing the US by train visiting prisons and promoting his "War On Crime." Stanley's idea was to have Cummings stop in Chicago and meet SAC Melvin Purvis and offer congratulations to him.
Cummings agreed and Stanley sent a memo to Hoover asking him to arrange to have the press at the train station to meet with Cummings. Stanley recommended Cummings meet with Purvis in Chicago; not with Hoover; not with Inspector Cowley. No one at the time had any idea of the immense publicity that would follow. At a desparate period in history, the American public was looking for a hero and found one in Purvis through media attention.
Documents from the Dillinger file reveal that on or about July 25, 1934 Purvis then traveled to Washington, D. C. at Hoover's request bringing with him Dillinger's gun, hat, glasses and other items that the press was clamoring to see. Hoover met him at the train station and the well known photo of them shaking hands was taken. At that time, sitting between Hoover and Cummings, he gave his first Washington press release on the events that unfolded days before.
The press interview is here. (enlarge the view if necessary)