One day before the federal government’s $349 billion aid program for small businesses was set to go live, the chief executive of a Minnesota bank was frantically dialing officials in Washington. Grand Rapids State Bank needed more time to understand the program, said the executive, Noah W. Wilcox, even as it faced a crush of borrowers. His pleas went unheeded.

“Somebody put a stake in the ground and it just wasn’t moving,” said Mr. Wilcox, who is also the chairman of the Independent Community Bankers of America, which represents about 5,000 institutions. “Secretary Mnuchin was not budging one inch from the date he initially set.”

Late on April 2, just hours before the opening, the Treasury Department released details on the Paycheck Protection Program. Treasury Secretary Steven Mnuchin told would-be borrowers that they would receive funds within a day, but the program, which promised speed, also brought chaos.

The rules confused lenders, including small community banks as well as Wall Street firms less familiar with the Small Business Administration, which was establishing the program. They were unsure about who would qualify for loans, how the loans would be distributed and how they would eventually be forgiven.

Some of the program’s rules were lax, allowing more than 200 publicly traded companies to obtain loans totaling more than $750 million. Yet, some small businesses that did get loans called the rules too restrictive, saying that it would be more helpful if they could use the funds to retool their operations for after the pandemic rather than paying employees.

The government has since published new guidance strongly discouraging public companies from using the program and urged those that did take the money to return it. Some have; others haven’t.

And the cash ran out in 13 days, leaving many small business owners waiting in line and increasingly desperate. As the federal government prepares to replenish the fund with $310 billion more, lenders expect the second round, which will open for applications at 10:30 a.m. on Monday, to be depleted even faster.

“As soon as they turn the switch on, that money will be gone,” said Tony Wilkinson, the chief executive of the National Association of Government Guaranteed Lenders, a trade group.

Small companies — those with under 500 workers — employ nearly half of America’s private sector work force. Most run on thin margins and have scant savings. For restaurants, gyms and other small businesses that depend entirely on people walking in the door, sales fell to zero after stay-at-home orders went into effect.

The Paycheck Protection Program was intended as a backstop to dissuade layoffs. Under the program, small businesses could borrow up to two and a half times their average monthly payroll cost. If they used the money to retain workers and keep paying them for at least eight weeks, the loan would be forgiven in full, and they could use a portion of the cash for certain other expenses, like rent and utilities.

“We knew demand would be through the roof,” said Jim Donnelly, the chief commercial officer at Bangor Savings Bank in Maine.

The program was hastily designed after a difficult debate in Congress. Democrats had wanted the government to provide cash infusions to small businesses through tax rebates or by having the Treasury work directly with payroll processors to administer payments. They had also wanted it to cover 16 weeks of payroll, rather than eight. Republicans wanted to steer the program through private sector financial institutions. They won.

The government adapted a decades-old program run by the S.B.A. that guaranteed small business loans issued by a network of banks nationwide. That allowed the government to essentially outsource most of the program’s legwork to banks, making its adoption speedier. But the S.B.A. and the banks had never before operated a program of this scale. Last year, the S.B.A. backed around $30 billion in loans. Now, it was expected to process more than 10 times that volume, in just a few weeks.

And the administration wanted the money out the door immediately beginning on Friday, April 3. “This will be up and running tomorrow,” Mr. Mnuchin said on Thursday night. “You get the money. You’ll get it the same day.”

To bankers, that was an absurd promise. It typically took them days to prepare even the simplest loan paperwork and go over the fine print with borrowers. As Mr. Mnuchin spoke, they were still waiting for critical information and materials. For example, the S.B.A. usually required lenders to use a specific promissory note for loans it guaranteed. The agency had not yet provided that note for the new program’s loans.

Many community banks decided to simply start lending and accept the risk that some details would need to be hashed out later. BancFirst, Oklahoma’s largest S.B.A. lender, set up a war room, enlisted employees from across the bank, and ran six-hour work shifts around the clock starting Friday night.

“We went 24/7 for four consecutive days,” said Kent Faison, the president of BancFirst’s commercial capital group. The volume was a “fire hose” like he had never seen before. Last year, his group made around 130 S.B.A. loans. This month, it issued more than 5,500.

Larger banks struggled to handle the staggering customer demand, especially as bank divisions that don’t typically deal with the S.B.A. were pulled in. Banks also imposed their own rules on who to lend to.

Bank of America started accepting applications right away, but its rules blocked many of its customers. JPMorgan Chase’s customers found themselves trapped in an enormous backlog. Wells Fargo, constrained by lending restrictions imposed by the Federal Reserve for its history of bad behavior, told most of its applicants that it would not be able to help them. And Citibank waited days to even begin taking applications from most of its small business customers.

As borrowers’ applications flooded in, gaps became apparent. Under longstanding S.B.A. rules, landlords, money lenders, political lobbyists and some other businesses were excluded. So were felons convicted within the last five years — an exclusion that cut off some people trying to rebuild their lives, and the employees who worked for them.

Other companies that seemed to test the program’s boundaries slipped through. In addition to several big restaurant chains, mining companies, drugmakers, software developers and manufacturers with access to equity markets and commercial loans were able to get loans using the taxpayer-funded program.

For some small businesses, the program worked. In late March, sales for Thomas Fennell’s translation business in Omaha, C3 Translators, dried up. He feared he would soon run out of cash to keep paying salaries for himself and his husband, the company’s two employees.

Mr. Fennell applied for a loan through the First National Bank of Omaha two days after the program opened. A week later, his application was approved. His check, for just under $20,000, arrived on April 17.

Because the loan will cover payroll, he plans to spend some of his savings to hire a local company to build a website for his business.

“I feel a strong moral obligation to keep this money circulating in the economy,” he said.

But even a successful loan application can pose challenges.

Naomi Pomeroy has two restaurants in Portland, Ore.: Beast, a 26-seat establishment serving a six-course tasting menu at two communal tables, and, across the street, Expatriate, a high-end cocktail bar. On Thursday, she got an email saying that her application for a loan for one of the two restaurants had been approved.

The news would have thrilled thousands of other business owners. Ms. Pomeroy greeted it with grim reserve.

“I hesitate to take the money,” she said.

To have the loan fully forgiven, she would have to rehire her employees immediately and pay them for two months, even though she cannot yet reopen her restaurant. The last of the 30 people she laid off five weeks earlier just started receiving unemployment benefits — and, with the $600 a week in additional federal benefits included in the stimulus bill, all of them are being paid more on unemployment than they earned working at her restaurants.

If Ms. Pomeroy takes the loan, she would prefer to spend the money retooling her spaces to fit social distancing guidelines. For Beast, that would mean getting rid of the two enormous tables and finding a way to turn a profit while serving fewer customers at a time. But renovations are not a permissible use of the emergency funds.

“If we could only open back up at 50 percent capacity and I was forced to hire back 100 percent of my staff, then that doesn’t make sense,” she said. Ms. Pomeroy, who helped found an advocacy group, the Independent Restaurant Coalition, said she was not sure the aid program would really benefit anyone in the restaurant industry.

Lawmakers and Treasury officials played down the program’s hitches. On Sunday, Mr. Mnuchin told Fox News that the first round “impacted about 30 million workers,” and that he expected the second round to support roughly the same number.

Lenders fear that many business owners will be left wanting, again, if the new funding runs out fast. Economists have estimated that more than $1 trillion might be needed to fully meet the program’s demand.

Bankers are preparing for a frenzy on Monday morning. Mr. Faison, at BancFirst, already has more than 850 applications ready to go.

Many lenders are telling applicants to brace for disappointment.

Robert A. Klein, who runs a management consulting business in Great Neck, N.Y., was one of thousands of Chase customers who got an email this week warning him that the bank had “an extremely large volume of applications” ahead of his.

“We expect that funds could run out again quickly,” Chase wrote. “We wanted to give you this information, so that you can decide if you would like to try applying with another lender.”

Jeanna Smialek contributed reporting.

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