I think he looks like George Skakel, but since he was dead at the time, maybe one of his brothers?
Ok. Mission accomplished. 
It’s hard to make out much from that photo, but could it be Jackie’s step-father, Hugh D. Auchincloss?
He’s on the stairs rehearsing for the grassy knoll.
No, the guy on the stairs is definitely older than Cox, has a heavier build, and looks very little like him.
No, I don’t think it’s him, either: http://i.dailymail.co.uk/i/pix/2015/06/18/21/29C18C8A00000578-3130135-image-a-47_1434661160006.jpg
I found a better quality image, but I still don’t know who he is.
Hmm, no, it’s not Spencer Tracy–he looked much more aged by late 1960 (and his head was less rectangular, more square). I think the unknown gentleman is an aide of some kind–the image on Getty identifies literally everyone else in the pic but him.
For the record, going from left to right: Bill Walton (journalist, painter & friend of Pres/Mrs Kennedy), Pierre Salinger, Ethel Kennedy, RFK, JFK, RFK’s secretary Angie Novello, and William F Haddad (Special Assistant to RFK at the time–and I believe of this group, the only one who’s still living as of January 25, 2018, so hypothetically you could try to get him to answer! 
).
(The man in question is only referred to as “Unidentified man on stairs.” FWIW, he does look like a Kennedy, but if he were, he’d surely have been named by photographer Jacques Lowe.)
I noticed that a copy’s in the NY Historical Society museum. If I remember correctly, there’s actually a Doper who knows a lot about history and might have some info… stuyguy, maybe?