By Leslie Josephs And Annie Gasparro 

Orders for organic burritos, Thai stir-fry and other frozen products from Amy's Kitchen Inc. have been growing so quickly that late last year the company bought a second factory--an Idaho plant H.J. Heinz Co. had just shut amid shrinking demand for its frozen food.

Started in 1988 by a married couple who named it after their daughter, Amy's in recent years has expanded beyond the world of specialty grocers into mainstream supermarkets like Piggly Wiggly and Price Chopper. Sales have surged 72% over the past five years to $443 million in 2014.

"Everyone said when we reached a certain size our growth would slow," said Andy Berliner, co-founder and chief executive. "We keep waiting for that to happen, but we're still growing so fast."

Amy's and other smaller companies focused on natural and organic foods are feasting on shifts in tastes among consumers distrustful of established food giants' products and ingredients. The rise of these smaller companies, helped by growing interest from big retailers, is eating into demand for brands that for decades were commonplace in American kitchens, like Kraft Foods Group Inc.'s macaroni and cheese and Kellogg Co.'s breakfast cereals.

The latest illustration of the tumult came Wednesday with the announcement that Heinz, two years after being taken over by Brazilian private-equity firm 3G Capital Partners LP, will acquire Kraft in a deal valued at around $49 billion. The hope is that the pair's many well-known brands, under 3G's penny-pinching management, will be able to revive growth.

The 25 biggest U.S. food and beverage companies collectively had 45.1% of the industry's $418 billion in 2014 sales, down 4.3 percentage points from five years earlier, according to Credit Suisse analyst Robert Moskow. Smaller brands' share rose to 35.3% from 32.1% in the period, with the rest going to companies that produce store-brands for big grocers.

Other small food companies have grown even faster than Amy's. Granola bar maker Kind LLC, started in 2004, has gone from less than 0.5% share of the snack-bar market in 2011 to nearly 6% today, according to Bernstein Research. Chobani Inc., whose Greek-style yogurt revolutionized that sector, reached $1 billion in sales its first five years of sales.

One big reason for these companies' success is retailers that long worked hand-in-glove with giant national brands--who spent big on advertising and paid for prime shelf space--have started seeking out upstarts to spice up their aisles.

Kroger Co., the biggest conventional grocery chain, with over $100 billion in sales last year, in 2013 started selling FlapJacked pancake mix from a small Colorado company, in its King Soopers chain there. After helping them develop a new package size, marketing strategy and additional flavors, Kroger now sells FlapJacked foods in more than 500 stores across the country.

"Our customers are increasingly telling us that buying local or buying from boutique producers is something they want, and we are working even harder to provide it," a Kroger spokesman said.

Last year, Bi-Lo Holdings LLC, a Southeastern grocery-store operator with about 800 stores, including the Winn-Dixie chain, launched two Shark Tank-like programs to offer local food makers a chance to pitch their products for space on its shelves.

"The food industry is changing at such a rapid pace, and we want to make sure we're not just keeping up with it but also driving it," said Nicole Hatfield, a senior manager at Bi-Lo Holdings.

More than a decade ago, Bonny Shuptrine and her husband began packaging her homemade spiced relish when patrons of their art gallery bought artwork. The reception was good enough that they founded Shuptrine's Twisted Products. Last year, Ms. Shuptrine presented samples to Bi-Lo executives at a supermarket in Chattanooga, Tenn. The chain put it in 23 stores, and Ms. Shuptrine has moved her production into an industrial kitchen.

Being big used to be an advantage, with a bigger marketing budget and buying supplies in bulk. Now it's a hurdle. Big companies are responding by acquiring the upstarts, or rolling out new products to try to compete with them. Kellogg recently came out with Origins cereal and granola, described as "real food prepared simply," while Campbell's launched a line of organic soups.

"We are well aware of the mounting distrust of Big Food," Campbell Soup Co. CEO Denise Morrison said last month.

Winning back consumer trust is proving hard for the incumbents. "The biggest challenge is that they are prisoners of their own heritage," said Mr. Moskow of Credit Suisse. "'Organic Velveeta' is a contradiction in terms, for example."

Smaller companies make up for their lack of financial resources with greater agility and a lack of historical baggage.

"New brands have a blank sheet of paper for telling their story. Established brands have to work against negative perceptions," said David Garfield, a consultant with AlixPartners.

Growth brings its own challenges for newcomers.

Alison Bailey Vercruysse, founder and chief executive of 18 Rabbits, a San Francisco-based maker of granola and granola bars, in 2011 tapped an outside food manufacturer to help her ramp up production. But the facility repeatedly failed to deliver her order, she said. She said her products had been in 80% of Whole Foods stores around the country, but the stumble knocked her down to 10%.

"Since we didn't have our own production capabilities, that almost killed us as a company," she said.

Now using a different producer, Ms. Vercruysse recently struck a deal with Target Corp. to sell an 18 Rabbits Jr. granola line targeted at children. "It took two years to bounce back" from the outsourcing stumble, she said. "2016 will be break-even."

Amy's, too, has had its headaches. This week it voluntarily recalled about 74,000 cases of lasagna, enchiladas and other meals with spinach due to listeria concerns with its supplier.

Amy's, based in Petaluma, Calif., now has about 2,000 employees who churn out more than 700,000 meals a day. Nevertheless, it boasts that its workers still stretch pizza dough by hand and don't use dehydrated vegetables. "Other companies manufacture food. We cook it," said Mr. Berliner, who runs it with his wife, Rachel.

The founders fight hard to keep that reputation. In September, when natural-foods peer Annie's Homegrown sold to General Mills Inc., Amy's got phone calls from concerned customers, prompting it to issue a news release headlined, "Which Mac & Cheese Brand Just Got Sold? It Wasn't Amy's!"

"If we were to sell to a big food company, our product would change, because they would want to improve profits," Ms. Berliner added.

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