Well, the 30-06 could not have been the first round for a rifle made in 1903, since it did not come along until 1906. First round was a round nosed, 220 gr load called the 30-03. (got 1 round in my collection- bit scarce)
And yes- the Colt Model 1903 .32 Smokeless Rimless was issued. In addition to certain used by OSS personnel, it was the official General Officer's sidearm. George Patton notwithstanding, when you made Brigader, you were issued a .32 with holster and belt, pistol engraved with our name. NRA museum in Fairfax has Eisenhower and MacAthur's .32s on display.
G'night kids- play nice, keep the TV down.
What we have here is... failure- to communicate.