The Chancellor should "show flexibility" and target companies that are still viable when the existing programme ends in October, members of the Treasury select committee said.
Critics have warned that in its current form - where it has been available to all companies, paying up to 80pc of employee wages if they are temporarily unable to work - furlough allows even failing "zombie" businesses to struggle on.
But the committee said the Chancellor should "carefully consider" whether a targeted extension was needed for furlough, which has protected 9.6m jobs so far at a cost of £35bn. The scheme will be wound up at the end of October.
Chairman Mel Stride said: "This requires a very difficult set of judgments; it is where careful analysis and creative thinking will be critical. As the committee has said throughout the crisis, the Chancellor must continue to show flexibility in his approach."
The report is likely to heap further pressure on Mr Sunak to change course, with business leaders and MPs across the political divide calling for a more targeted successor scheme aimed at sectors hardest hit by Covid-19, including the hospitality and entertainment industries.
It is feared the economy will be hit hard by a new ban on socialising in groups of more than six, enforced by "Covid marshals" with the power to snoop on pubs and restaurants. The new rules are likely to dent demand after a strong August in which sales at managed pub, bar and restaurant firms bounced back to 2019 levels due to Mr Sunak's Eat Out to Help Out subsidy scheme, according to data from the regular Coffer Peach Business Tracker.
Trade body UKHospitality estimates at least 350,000 jobs could go in the industry with a cliff-edge end to furlough, while other groups argue a slower tapering of the existing scheme could soften the blow this winter.
The MPs suggest that funding already announced by the Treasury could be shifted towards such a targeted extension, including the job retention bonus scheme, which hands companies £1,000 for every employee they retain up to January next year. The committee argues this will lead to "deadweight cost" because many of these employees would have been kept on anyway.
Echoing the committee's calls, Torsten Bell, chief executive of the Resolution Foundation think tank, said: "The Chancellor will need to reconsider his plans to swiftly phase out support given the painful reality that the economic crisis is here to stay."
The Government's fiscal watchdog, the Office for Budget Responsibility, forecasts unemployment rising to over 10pc next year, more than double the current 3.9pc.
Mike Cherry, of the Federation of Small Businesses, added: "The priority should be protecting viable small businesses - and all the jobs they provide - that have been disproportionately impacted by the coronavirus crisis, including those caught by local lockdowns, subject to continued national restrictions, or with staff that have directly suffered because of Covid."
Ministers have however announced more support for manufacturing after awarding the first £50m tranche of a £300m joint investment pot between government and industry to back hi-tech projects and bolster productivity.