Steve McNamara's side earned a hard-fought 32-26 victory over Samoa to open their campaign on Saturday, but Smith was forced to leave the Suncorp Stadium in Brisbane in a medical boot.
However, the Wigan Warriors man has moved to allay concerns about his fitness, insisting he will soon be back in training following a short recovery period.
"I've rolled my ankle, it's swollen up a bit and they've put the boot on as a precaution," he is quoted as saying by The Guardian. "It's just swelling.
"We've got to get the swelling down over the next couple of days and then it's back into training.
"I'll be back on the field on Monday. We've got a big week. We need to fix a few things."
There was also good news regarding Super League Man of Steel Daryl Clark, who came off relatively unscathed having been hurt in a heavy tackle during England's victory over Samoa.
"It wasn't a nice feeling when I was going back – you think the worst," he said. "After a couple of minutes, I realised it was nothing serious and I was pretty pleased.
"They say it's a bit of a rolled ankle and a dead calf. To come off that [tackle] with a dead leg, I’m pretty happy with that."
England are next in action on November 2 against Australia, who have never failed to reach the final in this competition but lost their opener to New Zealand on Saturday.