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“There’s talk of a multi-trillion-dollar program, given the size of the shutdown,” says Stephen Moore, a fellow at the conservative Heritage Foundation. “There’s a general recognition that we need something big to get some juice into the economy,” adds Mr. Moore, an outside economic consultant to the Trump administration and some congressional Republicans.

The ideas being floated include extending last week’s package to make the benefits last longer, as well as plugging in likely holes in the hastily assembled bill. One item in particular cited by both President Trump and Democratic leaders is a desire for more money to shore up state government budgets collapsing under lost tax revenues and new spending demands.

A common theme from economists and legislators across the political spectrum: The latest measure was mainly about keeping U.S. commerce on life support while it endures a medically induced coma. That is, paying businesses and workers revenues and wages lost during the shutdown. A next phase would likely pivot from stabilization to stimulus—providing the patient a robust regimen of physical therapy in an attempt to get the economy back to full health.

Action so far has been “about mitigation,” House Speaker Nancy Pelosi (D., Calif.) said at a Thursday press conference. “Next, we’ll go from emergency mitigation to recovery…to grow the economy and create more jobs.” She later called the new law “a very big down payment.”

As the coronavirus has ravaged U.S. business over the past month, lawmakers have reacted with uncharacteristic speed and unity to try to contain the damage. In just three weeks, a divided Congress has passed three major pieces of legislation to address the pandemic.

One question hanging over what is already being called “Phase Four” is whether that spirit of urgency and compromise can continue as the downturn advances. Or, will Washington return to the polarization that has often paralyzed Congress in recent years—especially as the November elections erode incentives for cross-party cooperation?

Another concern: Legislating amid travel restrictions and the risk that more in Congress come down with the disease.

Policy makers and economists will need to assess in coming weeks whether the most recent package does enough to tide over companies and workers through the end of the shutdown—whenever that occurs—or whether prolonged closures require another dosage of the same medicine.

The aid in the latest measure was calculated to help small businesses cover about eight weeks of payroll. That should be sufficient if the economy largely reopens by late May. But that is still far from certain. A longer quarantine might require, at a minimum, renewal of the same assistance.

“It wouldn’t make a lot of sense to keep the ecosystem of commerce alive for two months, just to kill it in nine weeks by not extending that program,” said Michael Strain, an economist at the right-leaning American Enterprise Institute, who has been working with GOP lawmakers on crisis-response measures.

Who Gets a Piece of the $2 Trillion Coronavirus Stimulus Package. President Trump signed a historic $2 trillion bill into law Friday aimed at relieving workers and businesses hurt by the new coronavirus pandemic. WSJ breaks down what is in the package. Photo: G. Ronald Lopez/Zuma Press

A bigger debate is emerging over whether Congress needs to do more, even if the disease crests by late spring. Some conservatives who supported the latest measure will likely be more hesitant about further action, worried that the crisis might turn into a permanent expansion of government intervention in the private sector.

“I would hope anybody that’s talking about a phase four would pause right now,” House Minority Leader Kevin McCarthy (R., Calif.) said Friday on Fox News. “Let’s make sure this is actually working in the process and be smart, get the data back of where—if—we do need more help.”

The main focus of Senate Majority Leader Mitch McConnell (R., Ky.) is on making sure the phase-three legislation is implemented properly and hasn’t yet pivoted to talks regarding future stimulus packages, according to a person familiar with his thinking.

Momentum might also be slowed by the second-guessing that inevitably follows big-spending programs, especially one thrown together so quickly. “My guess is that this bill won’t wear well over time, and Congress isn’t going to be inclined do another big package,” says Andy Laperriere, a Washington policy analyst with Cornerstone Macro, an investor advisory firm. “There will be fraud, companies getting money going into bankruptcy, things that people on the left and right won’t like.”

Any debate over new spending will likely reopen the ideological splits that were largely papered over during the past few weeks. “The left is going to want to do infrastructure, welfare payments and food stamps,” says Mr. Moore. “Our side will want to do tax cuts and deregulation.”

Mr. Moore has discussed with administration officials and congressional Republicans the idea of suspending the payroll tax through the end of the year, which he estimates would cost about $700 billion. Mr. Trump had pushed that idea early in the crisis, but withdrew it amid bipartisan objections on Capitol Hill.

The top agenda item for many Democrats is adding to the $150 billion the just-passed law doled out to state governments. Many analysts consider that insufficient. Democratic Gov. Andrew Cuomo branded the measure “really terrible for the state of New York,” saying the $4 billion his hard-hit state is slated to receive could be less than a third of the pandemic-induced budget shortfall.

“We don’t want states to say they can’t hire teachers or anybody else because they spent all their money on the coronavirus,” says Jay Shambaugh, director of the left-leaning Hamilton Project think tank and who was an economic adviser to President Obama.

At the signing ceremony for the latest bill, Mr. Trump on Friday also indicated there was White House support for further state aid “because the states have been hurt very badly.”

Mrs. Pelosi has said in recent days she would push for free health-care coverage for coronavirus patients and more money to shore up pension funds and food stamps. Many House Democrats want more funding for the District of Columbia, and the U.S. Postal Service. Former Vice President Joe Biden, the likely Democratic presidential nominee, has called for funds to forgive student-loan debts, as well as increased Social Security checks to aid at-risk seniors.

Given the uncertainty of forecasts—both around the spread of the disease and the state of the economy—many economists want Congress to implement so-called automatic stabilizers in a next bill: payments such as enhanced unemployment insurance that would immediately kick in if the economy remains weak, without requiring new legislative approval.

“The logistics of ramping up are much more complicated when you have to protect workers and consumers from a secondary outbreak,” says Diane Swonk, chief economist at Grant Thornton LLP.

Many simply want more time to analyze the problem and to craft a more thoroughly vetted response. “There’s real interest in the next phase being more deliberate, more thoughtful, more tailored,” says Steph Sterling, vice president for advocacy and policy at the left-leaning Roosevelt Institute, who has been consulting with Democrats on Capitol Hill.

“There’s a recognition of the optics of the scramble,” Ms. Sterling says, “and for more real analysis of what’s needed next.”

—Kate Davidson contributed to this article.