Ryder swapped with Williamson for the win in Hobart, making an unlucky duck - lbw despite strong suggestions of an inside edge - and 16.
The position has had a host of residents since the retirement of former captain Stephen Fleming in 2008.
Excluding one-off nightwatchmen Kyle Mills, Gareth Hopkins and Bracewell, 10 players have gone in at the fall of the first wicket in the 28 tests since Fleming walked away after the home series against England three years ago.
In order, they are James Marshall, Brendon McCullum, Ryder, Daniel Flynn, Jamie How, Ross Taylor, Martin Guptill, Peter Ingram, Mathew Sinclair and BJ Watling.
Fleming spent 43 of his 111 tests at No 3. His overall test average was 40.06, but at No 3, Fleming got his runs at 47.25.
By contrast the combined average of those tried since then is just 24.82. Nine half centuries have been put up, with four from lefthander Flynn the most by one player. Guptill has two, McCullum, Ryder and Williamson one apiece.
Flynn's average at No 3, from 12 innings, is just 29. Ryder is going at 28 from five tests in the job; Williamson is at 34.4 from three tests. Opener Martin Guptill has averaged 35 from his five times in the role.
Evidently Williamson is keen on the job long term while working out Ryder's best place in the scheme of things is also a work in progress.
Wright conceded after the second test that the No 3 job is "possibly" up in the air for the rest of the test summer.
"But we need to look at the batting order and makeup," he said.
"There is an expectation that the batsmen are on notice. What we have to get through is a simple message: your top six are expected to score runs.
"We won a test without scoring 300 runs in either innings at Hobart.
"You don't do that very often and we can't expect to win tests unless we achieve that."
Hobart was Wright's fifth test since taking charge at the end of last year.
But with the seam bowling looking well stocked, his attention will be firmly on the batsmen.
He has already spelled out what he wants to see from them and will doubtless be hammering that point home.
"What I'm looking for is substance in our batters, the ability to bat for long periods of time and sell your wicket dearly," he said.
"All these players are talented, and can play shots, but we need to value spending time at the wicket."
Cricket Australia will return to the voting of an expert to decide the man-of-the-match award following the controversy over Monday's Hobart test.
Mobile phone and tablet app-users shunned New Zealand quick Doug Bracewell's match figures of 9-60 and voted Australia's David Warner as best player for his unbeaten 123 in the losing side's second innings.
Bracewell polled just 27 per cent of the vote to Warner's 58 per cent.
"We were always running a bit of a trial for these two tests with those man-of-the-match awards," said CA chief James Sutherland.
New Zealand assistant coach Trent Woodhill described the decision as embarrassing.
