In the seven weeks since he moved across from the State Department to become White House Chief of Staff, Mr Baker has kept the lowest of profiles, rarely sighted and - until he granted a couple of interviews immediately after the first presidential debate on Sunday - never heard in public.
But with Mr Bush's poll standings telling their own dismal story, all that is to change. First the President blurted out during the debate that if he is re-elected his old friend will stay on to take overall charge of domestic and economic policy. Now Mr Baker himself is to make an important speech on the economy later this week.
In the meantime, the White House has confirmed officially what has been an open secret since the Republican Convention in August: that a second term will see an entirely new economic team, to replace Richard Darman, the budget director, Nicholas Brady, the treasury secretary and Michael Boskin, head of the council of economic advisers. Underlining the intention to 'clean house', Mr Baker yesterday sent out a letter demanding the pro forma resignation of all senior administration appointees, dated from election day.
But it remains to be seen whether these moves will suffice to win round an electorate whose faith in Mr Bush's economic credentials is next to nil. Bill Clinton, the Democratic challenger, has already delivered the obvious retort: 'They've had a losing season and the coach wants to fire the team. In America when you have a losing season, the coach gets fired, not the team.'
Equally pertinently and with less partisanship, another question is being asked: If his economic policies are correct and matters are nowhere near as bad as imagined - as Mr Bush again insisted on Sunday - why should his chief policy-makers be swept out, lock, stock and barrel? To that the President has thus far provided no answer.
He is, however, continuing to savage Mr Clinton on his alleged Vietnam draft-dodging and organisation of rallies against the war while a Rhodes Scholar at Oxford. The latest broadside was delivered at 8.30am yesterday, when Mr Bush put in an unscripted, but deliberately calculated appearance live on the NBC Today show, during what was to have been a segment on the White House staterooms, with Barbara Bush acting as tour guide.
Yet again he insisted that Mr Clinton had not put his record wholly on the table. 'I think that's an issue. It's not an issue of patriotism, it's an issue of character.' Indeed, the Republican camp has evidently decided the 'character' question remains the Democrat's Achilles' heel, despite Mr Clinton's double-digit lead in the polls and every sign that voters at large are infinitely more concerned with recession in 1992 than with what may or may not have happened in Oxford 23 years ago.
His remarks were plainly a curtain-raiser for last night's vice- presidential debate between Dan Quayle, Al Gore, and Ross Perot's running mate, Admiral James Stockdale (retd).
Having avoided the draft by opting for the National Guard in Indiana, Mr Quayle is not exactly invulnerable on matters pertaining to Vietnam. He was none the less expected to hammer Mr Clinton on the subject, amid lingering speculation he might have some unexpected new revelation up his sleeve.