Activist investor Ancora Advisors on Wednesday condemned J. Alexander's Holdings' board for rejecting its takeover offer and threatened to wage a campaign against the reelection of its directors later this year.

Ancora Chief Executive Officer Frederick DiSanto reiterated his disappointment in the company's leadership in an emailed letter to the company, arguing that its rejection of the activist's buyout bid is yet another example of the board's "inane" decision-making.

"The Board's response letter was nothing but spin to try to save face, and had zero substance on how you intend to create value for shareholders," DiSanto wrote in an emailed letter to the company's board.

"At the end of the day, do we want to fight them? No. But are we going to stick around? Yes," DiSanto told CNBC. "They are a good restaurant, they have good people. No one has said that the product they're producing is bad. It's about getting a rejuvenated board with a sense of urgency to continue to create value."

DiSanto said both in his letter and to CNBC that Ancora could try to rally other stakeholders to vote against the company's directors that are up for reelection if J. Alexander's refuses to explore strategic options.

The activist investor's second letter comes one week after it announced its intention to privatize J. Alexander's, a Nashville-based restaurant chain. Ancora, which manages about $6.5 billion from its headquarters in Cleveland, Ohio, offered on April 8 to buy the company for $11.75 per share in cash, or $186 million.

That figure represented a 24% premium to the share price when Ancora first disclosed its stake on March 12. The activist investor already owns about 1.3 million J. Alexander's shares.

But the $11.75 per share offer, according to J. Alexander's board, "is simply too unattractive to entertain and unanimously believes that to do so would not be consistent with the Board's fiduciary duties."

J. ALexander's Chief Financial Officer Mark Parkey did not immediately respond to CNBC's request for comment.

The company rejected the Ancora offer on Apr. 11, telling Ancora that its plan lacks "specific or verifiable details regarding [its] financing."

The board cited in its rejection letter that Ancora's bid is more than 12% lower than the company's 52-week intraday high of $13.40, and nearly 22% lower than the prevailing equity analyst price target. The one analyst that covers J. Alexander's has a $15 price target on the stock, according to FactSet.

The board "believes that it would not be a prudent or appropriate exercise of those duties to sell the Company at a price that significantly undervalues the Company at a time when the business is performing well," J. Alexander's board wrote.

"Allowing Ancora, or any buyer, to acquire the Company at a bargain price in order to reap the benefits of recent substantial investments in new restaurants ... would deprive our long-term investors of a significant opportunity," the board concluded.

JAX shares are down more than 11% over the last 12 months and up 7.8% over the last two years. The stock fell nearly 2% on Wednesday.