The Savannah Republican wants to lift the levy from 37 cents to $1.37 per pack, adding to the $5 or so a pack most Georgians pay to puff.
Stephens has pushed the idea in the last two sessions of the General Assembly. He has strong support from Democrats, but little in his own party, which controls both chambers.
"I'm just going to have to work a little harder at getting it done," he said.
He estimates his plan would raise at least $400 million a year, but says it can leverage $1 billion more in federal funds.
It would devote proceeds to Medicaid, the federal-state program that funds treatment for lower-income patients.
That would qualify Georgia for federal matching funds - about two and a half times what it paid in, Stephens said.
That in turn, would free up state money for trauma centers that treat critically injured people, he added.
Georgia voters rejected a Nov. 2 ballot measure to fund the centers by increasing auto tag fees $10 a year.
It will be hard to find another way to finance the centers because Georgia already faces a projected budget shortfall of at least $1.5 billion.
Stephens said the state pays about $500 million a year to treat victims of smoking-related illness.
"I think it's a no-brainer," he said of his plan. "It's time to stop subsidizing health care on the backs of taxpayers as a result of one product."
About one in five Georgians smoke.
So far, legislative leaders in a position to make or break the proposal remain opposed.
Lt. Gov. Casey Cagle, who presides over the Senate, "does not support raising taxes on Georgia's citizens," said spokesman Ben Fry.
Fry added Cagle wants to consider the recommendations - due in January - of a state commission on tax reform.
Interviewed on Oct. 14, House Speaker David Ralston rejected higher cigarette taxes.
"Nothing has changed," Marshall Guest, a spokesman for Ralston, said Tuesday.
One thing that is changing is the governor.
Outgoing Gov. Sonny Perdue opposed Stephens' legislation, but Gov.-elect Nathan Deal has said he would sign such a bill.
"I would say these are the kinds of things we have to look at," Deal said last November.
Stephens said Deal reaffirmed that position in a recent conversation.
Sen. Buddy Carter, R-Pooler, said the idea "makes sense," but added he wants to review the details before agreeing to back it.
Local hospitals also voiced support.
"It deserves very, very serious consideration" especially if it can attract federal matching funds, said Paul Hinchey, president and CEO of St. Joseph's/Candler. "We need to commend Rep. Stephens for taking another swing at this."
Phillip S. Schaengold, Memorial Health president and CEO, also praised the proposal.
Schaengold said it could make money available for areas such as trauma services "from those who cost the health care system the most money."
"More importantly," he added, "by increasing the price of cigarettes, it will deter young people from smoking."
Foes have said it would encourage smokers in border towns such as Savannah to buy cigarettes in South Carolina.
But Hinchey said that potential incentive has shrunk. The Palmetto State's cigarette levy rose 50 cents a pack on July 1.