Smith is now eligible to captain his country again after overcoming a two-year leadership ban.
Cricket Australia had banned Smith from the leadership role for two years for his involvement in the 2018 ball-tampering saga but the suspension expired in March this year.
See Also: Steve Smith speaks on potential return to Australian captaincy
Tim Paine is currently leading the Australian Test side, while Aaron Finch is captaining in white-ball cricket.
"I see no reason why someone shouldn't have a second chance and if part of that is the captaincy for Steve Smith, if he's the right guy and the most obvious candidate, I can't see a reason why he shouldn't be allowed to do it," Gilchrist told Fox Cricket.
"If Cricket Australia and the selectors feel that way, and Steve Smith is keen to do it, I think they should make him vice-captain immediately.
"And whenever those opportunities come up when Painey (Paine) finishes, when Finchy (Finch) finishes he just assumes it like any normal, natural progression," Gilchrist added.
Earlier, Pat Cummins and Travis Head shared the vice-captaincy role but recently Australia named the former as the sole candidate for the job.
Gilchrist feels the future plan should be made clear and announcing Smith as vice-captain will put an end to all the uncertainty surrounding the subject.
"It takes out all the speculation and allows Steve to know, everyone to know, what the journey is going to be.
"It allows the natural progression to happen without the speculation. By doing that it means that when the captaincy changes there is not all this hype and uncertainty around it and focus being taken away from the cricket.
"It just means everyone is clear that's going to happen," Gilchrist said.
(With PTI inputs)