The Keane family sweetened its bid last week to take Mod-Pac Corp. private, but some shareholders aren’t sure just how sweet the new $29.6 million offer really is.

The Keanes – company chief executive Daniel G. Keane and his father, Kevin, Mod-Pac’s chairman – increased their bid for the company by more than $3 million, earning the unanimous endorsement of the three-member panel of independent directors that was evaluating the offer and approval from the Buffalo specialty printing company’s full board.

But some longtime Mod-Pac shareholders, including some who have been pushing the Keanes to take the company private since the mid-2000s, aren’t rushing to endorse the offer, largely because they say they can’t get a good handle on which Mod-Pac the Keanes want to buy.

Is it the Mod-Pac that was reeling for the second half of the last decade, as it struggled to replace the lucrative sales it lost when online printing company, VistaPrint Ltd., bought out its contract to send all of its printing business to Mod-Pac? The company whose sales didn’t budge from $48 million between 2007 and 2010, leading to losses during the first three of those years?

The company that, despite earning more than $3.1 million combined in 2010 and 2012, fell back in the red during the first two quarters of last year, losing more than $240,000 during the first half as sales turned flat and the profitability of those revenues weakened because of the make-up of those sales and higher paperboard costs.

For that Mod-Pac, the Keanes’ offer of $8.40 per share – a 17 percent premium from their initial bid of $7.20 per share in late October – doesn’t look too bad, especially when you consider that the company’s total sales are just shy of $60 million a year, a tiny sum for a publicly traded company.

But what if the Keanes are buying the other Mod-Pac? The one that earned a combined $2.7 million during the final two quarters of last year – just $400,000 less than the company earned in 2010 and 2011 combined. The one whose sales jumped by 11 percent during the second half and hit record highs in each quarter.

While Daniel Keane warned investors during February that they shouldn’t assume that the strong growth during the second half of last year will continue, investors like Brendan J. Walsh and Edward K. Duch are wondering if the Keanes aren’t trying to snap up Mod-Pac on the cheap, just as the company finally is getting back on track.

“I just think there’s a lot more value for shareholders to be unlocked here,” said Duch, a former Buffalo banker who now lives in Florida. “I think there’s just more to put on the table.”

The timing of the buyout bid, which initially offered to pay Mod-Pac shareholders a 31 percent premium for their stock, also raised eyebrows.

Two days after the bid was disclosed in late October, Mod-Pac reported that its business rebounded strongly over the summer, with third-quarter profits jumping by 43 percent as strong sales of its custom folding cartons propelled revenues to an all-time high. The fourth quarter was even better, with profits more than doubling to their strongest level in three years as sales hit set another record.

Walsh and fellow investor George F. Cole of Arlington, Va., have made their feelings known to the board committee that reviewed the Keanes’ offer, writing the second of two letters last week expressing their concerns that the initial bid was too low.

If the company’s earnings keep growing the way they did during the fourth quarter, they think the company could be worth a lot more, maybe even $15 to $20 a share.

Simply going private will do wonders for Mod-Pac’s profits, eliminating the $500,000 to $1 million that the company now has to spend to meet the accounting, reporting and legal requirements all publicy traded companies face.

“There’s a half million to $1 million in savings that goes straight to the bottom line when you’re not public,” Duch said.

Walsh and Cole think the Keanes’ dominant 41 percent voting stake in the company also may have stopped other potential suitors from thinking about making a bid of their own. But now that the Keanes agreed not to have their shares count in the upcoming merger vote, Duch wonders if other potential buyers might take a second look.

“I’m hoping this maybe gets some heads turned by venture capitalists or other printing companies,” he said.

And now, Mod-Pac’s fate is in the hands of the shareholders who aren’t members of the Keane family. The Keanes declined to comment.

“I can’t predict what the shareholders will do,” said Christopher Carosa, a Honeoye Falls money manager who runs the Bullfinch family of mutual funds. “Are they willing to fight for a higher price, or are they inclined to take the money and run?”