In addition to Hastie and Freund, Arthur Goldberg was in contention, but state supreme court justices Roger Traynor of California and Walter V. Schaefer of Illinois were also given some small consideration. I'm not aware of where Schaefer was philosophically but presumably he was acceptable enough to the White House - the negative point was supposedly that he was a Catholic who had converted. (This was a time when such religious considerations mattered politically) Traynor was viewed as too liberal, and certainly looks like he would have been substantially more liberal than White.
Ultimately though, White was the clear frontrunner and won out largely on the familiarity of the Kennedys with him, and the fact he was part of the administration, so had a lot of backers. So anyone qualified who they were familiar with and part of the administration in an ATL would likely have received strong consideration. Archibald Cox received minor consideration, as well as future AG - under Ford! - Edward Levi. Law Professor Soia Mentschikoff would be one of the first female names to crop up for Supreme Court consideration, but was never seriously considered.