TR's choice of Fairbanks in 1904 was one of relatively limited options (I recall from Morris' biography that TR said something like "who on earth else is there?").
Anyhow, I submit that TR and Hughes were both consummate professionals who could set aside personal differences to work toward a common goal or set of goals. Therefore, I see a TR elective term beginning in 1909 with Hughes as Secretary of State, Taft moving from War to Justice as Attorney General, and Philander Knox taking War. At the first opportunity, Taft will be appointed to a long-coveted Supreme Court seat: so much for a TR / Taft rift as in OTL. Herbert Hadley of Missouri is a not-unlikely candidate for Justice thereafter.
That sets the table for 1912: Hughes for president and Hadley for vice president. That's a ticket that should be at minimum acceptable to all but the extreme fringes of the party. I can't see at this point that the butterflies are far-reaching enough to have a Dem ticket other than Wilson and Marshall, so let's go with that. In a general election, Hughes doesn't have Wilson's gift for lofty but vague rhetoric; offsetting that, however, is a solid record as both governor of New York plus Secretary of State: in other words, he has all the tools to be president. TR will likely make a few speeches supporting Hughes (and Bryan will likely make a few supporting Wilson). It has the makings of a close election between two progressives, but I think Hughes would prevail, albeit narrowly.
When the Austro-Serbian crisis happens in late June 1914, look for TR to cable the White House saying in effect that now's your chance to do something particularly solid and consolidate America's place in the sun in the bargain: namely, offer to mediate the crisis. Hughes knows an opportunity when he sees it, and makes the proposal to the chancelleries of Europe (not merely the contesting parties, given the interlocking alliances)--and submits TR as the chief mediator given his track record in ending the Russo-Japanese War. Kaiser Wilhelm II seizes on this impulsively, citing the astuteness of "my friend Roosevelt", all but demanding the Austrians go along with the idea. There is considerable support for mediation in Great Britain and Russia as well; ultimately, a reluctant Austria complies.
In assembling the mediation team, Hughes names TR as the head, supported by vice president Hadley and senator Gilbert Hitchcock--the latter calculated deliberately to involve both sides of the aisle in reaching a solution.