The Australian appeared to be knocked out after a heavy collision with Leeds Rhinos hooker Kruise Leeming, seemingly ending his hopes of playing at Elland Road.
But Smith insisted that he did not know whether Coote had failed his HIA.
"I can't tell you," he said. "I'm not sure is the answer to that.
"We certainly weren't going to put him back out on the park after that. He's a concern after getting sparked.
"He's absolutely fine now so that's not a concern. What would be of concern is how he is for next week."
As well as Coote, Rovers lost Sam Wood and Matty Storton to injury in a damaging 12-0 defeat at Headingley.
Wood is unlikely to feature against Huddersfield but Smith has not given up hope of fielding Storton.
On Wood's knee injury, Smith said: "It's a concern. We don't think it's an ACL, it's another part of his knee. I'd be surprised if he's available for next week.
"Stoz has done his AC in his shoulder. Some people are able to play with them and some can't. We're not sure about next week."
Rovers were also left sweating over a potential ban for Albert Vete after he was sin-binned for a high tackle but Kane Linnett could play at Elland Road despite being ruled out for up to three months earlier this week.
"He's a bit of a surprise that he may be available to us after I told you he won't be," said Smith.
"He will be able to play on. He is of risk but the consultant has told him he can't operate on him for a couple of weeks.
"If it happens to tear entirely, he's going to repair it anyway. I think Kane will probably carry on and see what happens with the result next week."
The Robins were outenthused by Leeds on Friday night, a worrying performance eight days out from their most important game of the season.
Smith felt KR's Easter exertions caught up with them on a disappointing night at Headingley.
"We were flat from the start," he said.
"We messed up on two kicks and were sloppy in some of our execution.
"It can happen after Easter. The boys put in a big Easter. I'm not going to flog these boys by any means.
"It wasn't a good performance, I'm not shying away from that. We're not happy that's what we came up with.
"I won't say it was expected but it often happens. Some of the games you see after Easter aren't high quality.
"But Leeds were good. Well done to Leeds. I'm not making any excuses about the performance at all. They deserved it."