Woodward, who led England to World Cup glory in 2003, said that there were "serious questions to answer" after the team's mediocre run of just two wins in the tournament.
But while the failure to achieve such a feat is not the end of the world given Jones' plan to build for the next World Cup in Japan next year, a third straight loss - against Ireland this weekend - would ring alarm bells for the Australian.
Jones, meanwhile, has called up Zimbabwe-born Exeter flanker Don Armand for England's final game of the championship against Ireland on Saturday following injuries to Courtney Lawes and Nathan Hughes.
That put Ireland 10 points clear of the English in the table, requiring a bonus point win for the visitors in Paris to keep the title chase alive.
England hooker Jamie George also said sorting out their own deficiencies would take precedence over stopping Ireland from winning the Grand Slam. He has often spoken about how he dreams of coaching the flawless game. Well, even the ideal game will not win England the championship now.
This is now uncharted territory for Jones' side, given they had lost just one Test during his tenure heading into this championship.
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However, their decision to take points off the kicking tee over territory suggested they were not thinking about the four-try bonus point, a decision Woodward found baffling.
"I think targeting a player and saying that they're going to be putting an very bad lot of focus on them, that they're going to be going after them and shutting down their time.I have no issue with it provided it doesn't get personal and it's not angle-based in a nasty capacity".
"Significantly, we gave too many penalties away, which was the difference in the game, and the breakdown again caused us trouble".
A simple pass would have resulted in a Hogg try but Jones made a mess of it and the chance went begging. It is time he does not have, since Ireland will surely use exactly the same tactics at Twickenham on Saturday.
Speaking about what it means to the current crop of players to be crowned champions, Schmidt said: "From a wider squad perspective, it is particularly satisfying". He has brought through a handful of good new players who are shaping well for the World Cup next year.
It's a long way from Woodville in New Zealand to the bright lights of Twickenham in London, but If Joe Schmidt's Ireland team beat England on Saturday it will be the biggest milestone yet on a journey that began in a small town on the North Island. In front of the French posts, the England backs were poised for an overlap but a knock-on in a ruck finally ended the match and England's two-year reign.