The ISM service sector report, while just in line with expectations, was a bright spot after a parade of bad reports—like last week's soft ISM manufacturing data and personal consumption.  Those data points were viewed as negative by a stock market that has seen bad economic news as a bad omen for stocks because of the parallel negative impact on corporate earnings, expected to be negative for the first time in six years. The March jobs report of just 126,000 nonfarm payrolls showed across-the-board weakness and raised concerns that the softness could extend beyond the first quarter.  Peter Boockvar, chief market analyst at The Lindsey Group, said it appears the dollar was propping up stocks, as well as the expectation of an easy Fed.  Read MoreWeak jobs signals dovish Fed  "I can't fully explain how we're 25 points off the lows from Friday. The S&P is less than 2 percent off all-time highs in light of what's going on. The addiction to the Fed just won't end," he said.  Dudley said the economy is likely to speed up after the first quarter's sluggish growth of about 1 percent, a view shared by many Wall Street economists.  Read MoreThis is when to worry about the stock market: Tom Lee  Seen as a proxy for Fed Chair Janet Yellen, Dudley's comments carry more weight than other central bank presidents who have said recently that June was not off the table for the first rate hike. The markets, however, have pegged September or later as a more likely time frame for the Fed to move, particularly after the weak jobs report.  The ISM nonmanufacturing survey Monday showed the service sector expanding, but at a slightly slower pace than last month. Both the new orders and employment components rose.  The index came in at 56.5 in March compared with 56.9 in February. A reading above 50 signals expansion. The 10 a.m. EDT report was preceded by Markit's service-sector data which showed a sizeable improvement to 59.2 from 57.1, the highest since August.  Read MoreWeek Ahead: Fed in focus after weak jobs  JPMorgan economists point out that while the two service sector reports went in opposite directions, both look consistent with GDP growth above 2 percent, suggesting the slowdown in the first quarter is temporary. The economists said the reports make it appear the March employment report may also have overstated weakness. JPMorgan has said first-quarter growth was tracking at just 0.6 percent.  Treasury yields were at their highs of the day late morning. The 10-year was at 1.90 percent after falling below 1.80 briefly when the jobs number came out Friday.  "The fact the service sector indicators were solid is a bit of solace for those expecting the economy to grow at a reasonable clip," said Ian Lyngen, senior Treasury strategist at CRT Capital. "I think it was more an issue of people looking to opportunistically add to equity exposure following the pullback after the NFP (nonfarm payrolls) rather than something new this morning."  He said some investors clearly look at weaker data as an excuse for the Fed to stay easy longer, a positive for stocks.  O'Rourke said the stock market spring back could be temporary, and stocks could face new headwinds as the week progresses. The weak economic data have been a concern for stock traders, since earnings growth is expected to be negative for the first time in six years.  The Dow was up 117 points to 17,880, and the S&P was up 13 at 2,080. A six percent rally in U.S. oil futures also helped lift stocks as the energy sector gained. Other commodities, like gold, also benefited from dollar weakness during the trading day. By late in the day, the dollar reversed course and was trading higher.  "I think it's a tape with a lot of people still out. It's noise and the next week or so, we'll see how earnings shape up," said O'Rourke. "This market has largely been driven by policy. ... The risk is if earnings miss, stocks get more expensive not because prices are going up but because earnings are going down. That's what I'm watching. I think it will be hard for the market to make significant headway if earnings are not good because I think the policy peak has passed."
The promise of an easy Fed, a weaker dollar and service sector data that showed an expanding economy helped lift spirits and stocks Monday.