"If you get a sub-300,000 (claims) number, it will be all good," said Adrian Miller, GMP director of fixed income research. He added that the claims are being watched closely to see if jobs related-data is all on the same path of improvement.
The S&P 500 jumped 1.2 percent to 2025 Wednesday, erasing a good chunk of a 4.8 percent loss made between Dec. 29 and Jan. 6. The ADP report heightened expectations for an upside surprise in Friday's jobs report, which would be a positive catalyst after concerns about Europe and sinking oil prices soured the new year market mood.
Oil prices stabilized Wednesday, with West Texas Intermediate futures for February rising 1.5 percent to $48.65 per barrel, reversing overnight losses. Brent futures rose 0.1 percent to finish at $51.15 per barrel, after falling through $50 a barrel.
Data showing deflation in the euro zone Wednesday raised expectations that the European Central Bank would announce a quantitative easing program later in the month, and that weighed on the euro, sending it to a new 9-year low.
Read MoreStrategists' stock picks for 2015's economy
U.S. yields meanwhile rose, with the 10-year Treasury at 1.97 percent and the 30-year yielding 2.572. Shorter-term Treasurys were mixed late Wednesday after some bond investors viewed the Fed's minutes from its December meeting as slightly dovish.
Stocks have been volatile in the first days of January, and traders are watching to see if the market is higher or lower after its first five days, which end Thursday. The S&P 500 would have to reverse the 1.6 percent loss its seen so far year-to-date and then some to send a positive market signal.
According to the Stock Trader's Almanac, the last 41 positive first five days were followed by full-year gains 85.4 percent of the time, or 35 times. The average gain was 14 percent for the following year. Exceptions include 1994 and four years when there was war in both Vietnam and the Middle East.