Strangely, although the album notes present Norman Granz' information about the recording place (Syria Mosque Auditorium in Pittsburgh, PA), the mostly reliable jazzdisco.org site still offers a different place and time ("live in Los Angeles, CA, October 7, 1946"), see here.
Although that is not on my list (because I haven't found images of a copy yet), "How High The Moon" was possibly released earlier, on the first run of the Clef label in 1947, see there:: "How High the Moon" was Clef 107 (parts 1 and 2) and 108 (parts 3 and 4) and Merc/Clef 11009 (parts 1 and 4) and 11010 (parts 2 and 3)
As has been proved, Norman Granz' own first Clef label had released Volumes Six and Seven of the JATP albums, with individual discs 101 to 106, see the label images on my list. After that (and possibly after "How High The Moon"), Clef was shelved until it was revived in summer 1953.
"How High the Moon", the first tune on the first Jazz At The Philharmonic (JATP) volume in 1945, had become the theme song of JATP two years later. As the original recording was on the Asch Records label and the Stinson Trading Co. had acquired the rights and the masters (see the 1947 reissue on Stinson 453), it was not available for Norman Granz, the mastermind of JATP. Hence he substituted it for his series with an excellent new version featuring Coleman Hawkins, recorded on March 5, 1947 at the Syria Mosque Auditorium in Pittsburgh, PA.