By Steven C. Johnson and Lucy Hornby NEW YORK/BEIJING, Nov 1 (Reuters) - Private U.S. firms stepped up hiring last month and factories showed modest improvement, surveys showed Thursday, while Asia's large economies started to pick up after a year of slower growth. The jury was out on whether the data signalled sustained improvement in the fragile global economy, though analysts said strength in the United States and China, the world's two biggest economies, was essential to overall economic well-being. That's particularly so at a time when a debt crisis in the 17-country euro zone has plunged several countries in the region into recession. Reports on major euro zone countries are due on Friday and expected to show continued economic contraction. But the picture appeared to be brightening elsewhere. U.S. companies added 158,000 jobs in October, data from payrolls processor ADP showed on Thursday, far more than the 135,000 predicted in a Reuters poll. Separate data showed the number of Americans filing for first-time jobless benefits fell last week while U.S. consumer confidence jumped to a four-year high. In other reports, overall sales at major U.S. retail chain stores, excluding at drugstores, rose 4.7 percent in Octoer, compared with analysts' expectations for a 4.3 percent increase at the 17 chains tracked by Thomson Reuters I/B/E/S. But some U.S. retailers expect the massive storm that hit the U.S. East Coast this week to hurt sales in November, while overall holiday sales should still show the slow growth that was forecast before Sandy hit. The data was welcomed by the U.S. stock market, which rose on the second day since it reopened following a massive storm that battered the U.S. Northeast earlier this week. The data "are encouraging," said David Sloan, economist at 4Cast Ltd in New York. "There shouldn't be any distortions from the hurricane yet. It seems with the ADP message there is some evidence of labor market improvement. It is not totally convincing yet but overall the message is positive." The picture was more mixed among U.S. factories. The Institute for Supply Management said the pace of growth picked up slightly in the sector, with its index rising to a five-month peak of 51.7. But the pace of hiring in the sector slowed. A separate report from data firm Markit showed the slowest pace of growth in 37 months, the result of reduced demand for U.S. goods overseas. "It looks like manufacturing has stopped deteriorating. It's weak growth but it's growth," said Christopher Low, chief economist at FTN Financial. In Brazil, manufacturing expanded for the first time since March, according to the HSBC Purchasing Managers' Index, boosting hopes for economic improvement in the fourth quarter. ASIAN REBOUND Data from Asia was encouraging as well. China's economy, the motor of global growth in recent years, appears to have gathered pace in October after slowing to its weakest pace in more than three years in the third quarter. Chinese manufacturing showed renewed vim, with the official manufacturing purchasing managers' index rising to 50.2 from 49.8 in September. Economists said that could help lift fourth-quarter growth above the 7.4 percent rate recorded in the July-to-September period. Also on Thursday, the final reading of the Chinese HSBC PMI rose to 49.5 in October from 47.9 in September. The reading was the highest since February. The official PMI generally paints a rosier picture of the factory sector than the HSBC PMI as the official survey focuses on big, state-owned firms, while the HSBC survey targets smaller, private firms that have limited access to bank loans. "Overall sentiment is brightening and Chinese orders are suggesting a moderate recovery," said Hirokazu Yuihama, a senior strategist at Daiwa Securities in Tokyo. Beijing has been following a programme of pro-growth fine tuning of economic policies for a year and analysts broadly expect that to remain in place when a new leadership line-up at the top of the ruling Communist Party is unveiled this month. "The return of the PMI above 50 suggests economic momentum has indeed picked up. It indicates the effect of policy easing may have been stronger than the consensus expected," Zhiwei Zhang of Nomura said in a comment emailed to Reuters. "We believe macro data will continue to surprise on the upside in coming months, as the government continues to ease policy through the period of leadership transition." South Korea, another of Asia's manufacturing powerhouses, posted the first annual rise in exports in four months in October, adding to hopes for a turnaround after a year-long slump in global trade. Stronger demand from China and Europe led the small but meaningful 1.2 percent gain in exports last month on an annual basis, data from the Ministry of Knowledge Economy showed. There was less encouragement from South Korea's Markit/HSBC PMI, which showed the country's manufacturing sector shrank in October for a fifth consecutive month, although at a slower pace than in September. Taiwan also saw a continued manufacturing contraction, although the headline October reading from its HSBC PMI was the highest in four months. Monetary easing by the big developed world central banks has helped push up the currencies of countries such as Korea and Taiwan, hampering their export-led recoveries. India, hammered as much by internal politics as the global downturn, also showed signs of a modest revival in manufacturing. The HSBC manufacturing PMI, which gauges the business activity of India's factories but not its utilities, nudged up to 52.9 in October from 52.8 in September. The downturn in Britain's manufacturing sector, however, worsened in October as companies received fewer orders and costs rose at a faster pace, reviving worries about the country's fragile recovery.