The scrum-half made his senior debut against Zebre in January and has now signed his first professional contract with the Warriors, having previously been part of the Munster set-up.
His father, Greig, earned three caps for Scotland between 1987 and 1991 and was widely respected as a youth coach in both Scotland and Ireland. He died in a paragliding accident in South Africa in 2023 at the age of 58.
Smith says Oliver's upbringing in rugby is reflected in how he approaches the sport.
"I think he's a clever guy, understands the game well," Smith told BBC Scotland.
"You can see where his DNA comes from, it's all evident and obvious in the way that he prepares for training, he enjoys contributing - he's a rugby person through and through. And that obviously comes only if you grow up in that kind of environment.
"Fantastic to have signed him on. I think there's a young group stepping up and a lot of them will stay on and become pros, I believe, so Jack is one of the leading guys in that perspective.
"He's still young, he's still 22 years old, his body will still develop quite a bit in the next few years from a physicality perspective, but he's already got the skill-set and he's already got the mentality that's good enough to play at a higher level.
"So as he physically develops and gets better, I think he's got a good future ahead of him."
Oliver has been named among the replacements for Glasgow, who travel to face Connacht in the United Rugby Championship on Saturday.