New York's Dow Jones industrial index closed nearly 500 points higher on Tuesday, offsetting much of the 777 point fall the previous day after the House of Representatives voted against the plan by 228 votes to 205.
On Wednesday Japanese and Australian share prices opened sharply higher
following Wall Street's powerful rally overnight on hopes of a new US financial rescue plan.
Tokyo's Nikkei index rose 1.2 per cent to gain 136.75 points to 11,396.61 in the first minute of trading while Australia's benchmark S&P/ASX jumped 3.5 per cent, up 157.9 points at 4,758.4.
"Things are still very volatile and I think that the market is betting that there will be some kind of bailout plan ... and the moment they sense that there won't be they will go down significantly, and maybe then 777 points will look like small-fry," Al Jazeera's John Terrett, reporting from New York, said.
He said that there could be further volatility over the coming days as the House of Representatives was not due to return to Washington until Thursday, meaning there was unlikely to be another vote before the weekend.
Bush told taxpayers that he understood that many people were concerned about the $700bn cost of the rescue plan, but the cost of not passing the bill would be greater.
"That is a large amount of money ... but we are also dealing with a large problem, to put that in context the drop in the stock market yesterday represented more than $1 trillion in losses," he said.
He also said that he expected much of the money used to buy up the bad debts would be made back once they were sold as the market recovered.
"It is likely that many of the assets would go up in value over time, much, if not all, of the tax dollars invested over time would be paid back," he said.
Marsha Blackburn, a Republican who voted against the bill, said that many congressmen were still concerned about the level of taxpayer
commitment to the bailout.
"This is something we can't leave on the table and leave Washington. This is something that affects every man and woman, every family in this country," she told CBS television.
Al Jazeera's Rosiland Jordan, reporting from Washington DC, said the rejection of the bill had been a shocking loss for the Bush administration.
A substantial number of Republicans were saying they did not want the increase in spending and did not believe in any government intervention in the economy, our correspondent said.
The president on Tuesday appealed directly to those Republicans, especially the ones who opposed the plan as government interference in free markets, to get behind a deal.
"Our country is not facing a choice between government action and the smooth functioning of the free market. We're facing a choice between action and the real prospect of economic hardship for millions of Americans," he said.
Barack Obama, the Democrat presidential candidate, also gave a speech urging Congress to push ahead with attempts to pass a rescue plan.
"While there is plenty of blame to go around ... all of us have a responsibility to solve this crisis because it effects the financial wellbeing of every single American," he said.
"Its not a time to figure out how to take credit or where to lay blame, its not a time for politicians to concern themselves with the next election, its a time for all of us to concern ourselves with the future of the country we love."
But Tom Price, a Republican who voted against the bailout, said that rather than not acting he thought "it's important to get this vote right, not necessarily to get it quick".
"We are principled in the fact that we believe we ought to stick to American principles, we ought to protect the taxpayer, we need to make sure that private money, private equity can get involved and have Wall Street bail out Wall Street, not on the backs of the taxpayers," he said on NBC television.
Marcy Kaptur, a Democrat Representative, said: "We have to look for something that will make the markets function in the way that they should, not reward bad behaviour."
David Buick from BCG Partners in London told Al Jazeera that now was the time for pragmatism rather than recriminations.
"The fact is we have a problem and I don't think that those people who voted against it know the implications," he said.
"The banking sector is the lifeline, the best artery ... to the fabric of every society in the world. Without strong banks we have nothing, we have a delapidated economy worldwide, we have no retail, we have unemployment, we have anarchy."
European shares saw some rebound on Tuesday as investors became more confident that the bailout plan would be revived and eventually agreed.
London's FTSE 100 closed at 4,902.5 points, up 83.7 or 1.74 per cent while the FTSEurofirst 300 index of top European shares ended 1.6 per cent higher at 1,063.65 points. The index was still down 11 per cent of its value over the month.
Germany's DAX and the CAC-40 in Paris also ended the day up.
Earlier, reaction in Asia had been more mixed with Japanese stocks sliding 4.1 per cent to hit a three-year closing low and shares on Australia's S&P/ASX 200 dropping 4.3 per cent to their lowest close since December 2005.
But Hong Kong's Hang Seng recouped early loses to end the day 0.8 per cent higher after two days of sliding.