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UPDATE 7-Oil falls more than 2 pct on signs of higher output

* Brent posted biggest third-quarter increase since 2004

* U.S. drillers add oil rigs for first week since mid-August

* OPEC oil output rises in September - Reuters survey

* Libya's Sharara oil field under force majeure-sources (Updates throughout, adds quotes, adds context, changes byline, changes dateline from LONDON)

NEW YORK, Oct 2 (Reuters) - Oil fell more than $1 a barrel to below $56 on Monday as a rise in U.S. drilling and higher OPEC output put the brakes on a rally that helped prices to register their biggest third-quarter gain in 13 years.

Iraq announced its exports rose slightly in September while OPEC overall boosted output, a Reuters survey showed.

In its report on Friday, General Electric Co's Baker Hughes energy services firm said drillers added six oil rigs in the week to Sept. 29, bringing the total count up to 750.

"Weve seen them add rigs for the first time in seven weeks, so that changes sentiment as well," said John Tjornehoj, energy market analyst at CHS Hedging.

Brent crude, the global benchmark, was down $1.22 or 2.15 percent at $55.56 a barrel at 11:20 a.m. EDT (1630 GMT). It notched a third-quarter gain of about 20 percent, the biggest third-quarter increase since 2004 and traded as high as $59.49 last week.

U.S. crude was down $1.56 or 3 percent at $50.11. The U.S. benchmark posted its strongest quarterly gain since the second quarter of 2016.

The rally was driven by mounting signs that a three-year supply glut is easing, helped by a production cut deal among global producers led by the Organization of the Petroleum Exporting Countries.

But a Reuters survey on Friday found OPEC oil output rose last month, mostly because of higher supplies from Iraq and also from Libya, an OPEC member exempt from cutting output.

However, in a Monday letter the National Oil Company declared force majeure on deliveries from Sharara, the country's largest oilfield.

Middle Eastern oil producers are concerned the price rise will only stir U.S. shale producers into more drilling and push prices lower again. Key OPEC producers consider a price above $60 as encouraging too much shale output.

In February oil industry sources said Saudi Arabia would like to see oil around that $60 level.

Petromatrix strategist Olivier Jakob said Brent's weekly chart had formed a "shooting star", a pattern seen as indicating a market has reached a top.

Hedge funds have accumulated a record bullish position in middle distillates such as diesel, heating oil and gasoil, anticipating stocks will be relatively tight this winter.

"We've seen a run up in heating oil futures, and I think that particular product has supported the rise of WTI," said Tjornehoj, while noting that distillate prices fell on Monday. "As we reverse here lower we see the recent strong correlation continuing."

(Additional reporting by Alex Lawler in London and Aaron Sheldrick; editing by Susan Thomas; Editing by Edmund Blair and David Goodman)