For the insider story, read a book that, while not flawless (we tried hard for flawless, but didn't quite get there), was the first ever to trace the three-legged Vanguard-Sputnik-Explorer race - and the first to discuss the fourth competitor, NOTSNIK, aka Project Pilot.
Erika Lishock and I talked to all the leaders of the U.S. programs we could locate. Most helpful were James Van Allen, JPL head Bill Pickering, Ernst Stuhlinger of the von Braun team, and Milt Rosen, Technical Director of Project Vanguard. (All are gone now except Rosen: I talked last week to Pete Wilhelm of the Naval Research Laboratory, who told me Milt is still alive but no longer seeing people.)
So check out The First Space Race!