This begs the question, "Was there ever an unsuccessful test prior to Trinity?"
And this begs the reply.
Yes...there was an unsuccessful test...
.....and the resulting political mutation and ensuing social morphing resulted in this:
No. And I don't believe there was a "third bomb", either -- despite Tibbetts' reference to it.
All the historical references are quite clear. There was enough material for three "devices" -- the plutonium device that was tested at Alamogordo, the uranium bomb that was dropped on Hiroshima (everybody was certain it would work, no need to test it) and the plutonium bomb dropped on Nagasaki.
When Hiroshima and Nagasaki were obliterated, Hanford Works had not yet manufactured enough plutonium for a "third bomb", as Tibbetts described it. It was still several months away.
P.S. LeMay would've known this. In fact, did know it...
No