That would be an 11% premium over Midwest Air's closing price of $13.51 on Friday on the American Stock Exchange.
AirTran made its first proposal in October 2006 for $290 million in cash and stock. The offer - and a previous one - were first disclosed on Dec. 13. AirTran raised its offer to $345 million in January.
"We firmly believe in the underlying value and benefits of combining these companies, and we are committed to bringing together these two airlines to create a truly national airline well-positioned for success in an increasingly competitive environment," Joe Leonard, AirTran's chairman and chief executive officer, said in a statement.
Some Midwest passengers, shareholders and Midwest Air Group CEO Tim Hoeksema have wondered whether such a deal would end Midwest's traditions of wide leather seats and fresh-baked cookies.
But Leonard has said the merger would combine the best of both airlines, including the traditional Midwest perks.
The expiration date of the latest offer is May 16 and could set up a showdown the following week when Midwest holds its annual shareholders meeting on May 23 if the board rebuffs the latest overture. AirTran's shareholders meeting is slated for the same day.
"Midwest shareholders will have the opportunity, at that time, to control their own destiny by determining whether the existing board of directors is really serving the best interests of shareholders," Leonard said.
Leonard has said a combined company would reach $3.5 billion in revenues and have some 15,000 employees by the end of this year.