The federal government has piled up debt since the latest budget deal was signed into law, tacking $462 billion onto the national credit card since Nov. 2 as the Treasury Department replenished its funds and began another round of borrowing to take it all the way into 2017.

A staggering $339 billion in total debt was added on just the first day after President Obama signed the budget agreement — the single largest hike in history.

The debt has continued to rise, albeit more slowly, in the days since, putting the president on track to come close to the $20 trillion mark by the time he leaves office in January 2017.

Meanwhile, the early deficit numbers for fiscal year 2016, which began Oct. 1, are already looking more grim.

The government ran a deficit of $136 billion last month, up 12 percent compared with the previous October, as spending ballooned and taxes remained nearly flat. It was the worst October since 2010, when the government was still spending on the stimulus and was on pace for a deficit of more than $1 trillion that year.

The Treasury Department did not respond to a request for comment on the debt spike, but analysts said it wasn’t unexpected.

“It’s not going to keep rising at that pace. It’s like putting a cap on a geyser. It was being held at an artificially low pace,” said Robert L. Bixby, executive director of the Concord Coalition, which pushes for policymakers to control debt and deficits. “It’ll increase at a more traditional level from this point on.”

Despite the massive spread of red ink, the government has been getting away with small debt service payments because of historically low interest rates over the past several years.

But as rates rise, so will those payments — from about $220 billion a year now to $755 billion a year in a decade.

The size of the debt has begun to take a starring role in the 2016 presidential campaigns. In the Republican debate Tuesday, Fox Business Network prodded candidates on their plans.

Sen. Rand Paul of Kentucky poked fellow candidate Sen. Marco Rubio for a tax and defense spending plan that Mr. Paul said would hike deficits by $1 trillion.

“As we go further and further into debt, we become less and less safe. This is the most important thing we’re going to talk about tonight,” Mr. Paul said. “Can you be for unlimited military spending, and say, ‘Oh, I’m going to make the country safe?’ No, we need a safe country, but, you know, we spend more on our military than the next 10 countries combined.”

Mr. Rubio said defense comes first.

“We can’t even have an economy if we’re not safe,” he said.

Ohio Gov. John Kasich, who as chairman of the House Budget Committee in the late 1990s helped write the deals that produced four years of surpluses, said he has plans to do it again — including a freeze on non-defense discretionary spending.

But it was just such a freeze that Congress rejected this year, forcing the budget deal that allowed for unlimited borrowing for another 16 months.

Mr. Bixby said Congress should use those months to work on long-term fixes rather than preparing for another knock-down fight over the debt limit.

“The way to keep the debt from going up is to change the policies producing the debt,” he said.

The government began bumping up against the debt limit in March and was borrowing from other funds — using “extraordinary measures” — to keep from breaching the $18.1 trillion level. Treasury Secretary Jacob Lew was able to stretch that borrowing through the end of October, when Congress passed a debt holiday lasting into March 2017, allowing him to borrow as much as needed to keep the federal government operating.

The first move was to replenish all of the funds depleted under the “extraordinary measures,” which is what sent debt skyrocketing on Nov. 2.

Such spikes are normal. In 2013, when a debt deal was reached, the government added $328 billion to its borrowing in one day. After the August 2011 debt deal, the amount rose $238 billion in one day.

But the Nov. 2 spike topped them all, at $339 billion in one day.

Of that, about $199 billion is public debt, which is money borrowed from outside sources, and $140 billion is borrowing from within government accounts.

As of Monday, the gross total debt stood at $18.6 trillion, with $13.4 trillion of that public debt borrowed from the outside.

When Mr. Obama took office in 2009, total debt stood at $10.6 trillion.