One concern will be a potential tournament-ending injury to scrum-half Ben Youngs, who was forced from the field after 20 minutes as his knee swivelled against him when passing from the base of a ruck.
Youngs was carted off on a stretcher less than 10 minutes into his 56th start for England, which overtook Matt Dawson's record of most starts for an England scrum-half.
"We'll know more tomorrow but he's unlikely to be fit for the Wales game", Jones said immediately after the match.
Saracens' Richard Wigglesworth, who won the last of his 27 caps at the 2015 Rugby World Cup, and Wasps Dan Robson are possible call-ups with Gloucester's Ben Vellacott - a former Scotland under-20 global - also believed to be in contention.
Aussie Jones has always admired the hole-punching ability of rugby league convert Te'o, who was born and raised in Auckland before shifting to Australia in his late teens.
Looked a little out of sorts after time out because of injury.
Having risen to prominence in the Premiership semi-finals last May, before making his worldwide debut last November, Simmons has been trusted by Eddie Jones to start the Six Nations as his No 8 due to the injuries sustained by Billy Vunipola and Nathan Hughes, and the Exeter Chief certainly rewarded the head coach for his faith with two second-half tries and a performance of worthy of the global stage.
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England then capitalised when No 8 Simmonds, only starting because of injuries to Billy Vunipola and Nathan Hughes, marked his Six Nations debut with a try when he burst through yet more poor tackling.
Exeter Chiefs back row forward Sam Simmonds scored two tries on his tournament debut and Jones was full of praise for the number eight.
However, it was far from a masterpiece and one which will have left head coach Eddie Jones with plenty to ponder ahead of bigger challenges over the next two months.
'If he was a cricketer he'd be playing Twenty20. "It was a tough game of rugby but we ran away with it in the end".
Italy is hoping not to finish bottom of the table for the third successive year.
"We let them into the game a bit with individual discipline lapses but, in general, everything was positive".
England scored four tries in the final half an hour with Sam Simmonds scoring his first just moments after Tommaso Boni had a try disallowed that would have brought Italy to within three points with a conversion.
"I was really pleased with the effort". "We played against a great side, No 2 in world, and we caused them a lot of problems with the ball".