Midwest's shares closed down 4 cents at $13.96 on the American Stock Exchange on Tuesday. In after-hours trading, the stock gained $1.04 at $15.
The AirTran move came after a major Midwest shareholder objected to a proposed buyout led by private equity firm TPG Capital.
The increase to $16.25 per share from $15.75 marked the fourth time AirTran has raised its offer in its eight-month pursuit for Midwest, in the face of strong resistance from Midwest's management.
On Sunday, Midwest accepted a $16-per-share buyout offer from TPG Capital, backed by Northwest Airlines.
Midwest did not immediately return a call seeking comment on AirTran's newest offer.
AirTran said it raised its offer at the request of certain Midwest shareholders, which it did not name.
Earlier Tuesday, Pequot Capital Management, a large shareholder in Midwest, said it had "significant concerns" about Midwest's decision to pursue the TPG Capital deal.
Last month, another large institutional investor in Midwest, Octavian Global Partners, encouraged the airline to enter talks with AirTran.
AirTran said its latest offer consisted of $10 in cash plus 0.6056 of a share in AirTran, equaling $16.25 per Midwest share.