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UPDATE 4-Oil prices dip on doubts that recent rally will last

* Q3 rally pushed crude prices to mid-2015 highs

* Analysts say oil markets have largely rebalanced

* U.S. rig count suggests further rise in shale output (Updates throughout, changes dateline, previous SINGAPORE)

LONDON, Oct 4 (Reuters) - Oil prices fell on Wednesday, pulled down by caution that a rally that lasted for most of the third quarter would not extend through the last three months of the year.

U.S. West Texas Intermediate crude futures were at $50.10 per barrel at 0843 GMT, down 32 cents from their last close. They fell below $50 earlier in the session.

Brent crude futures were down 47 cents at $55.53 a barrel.

The fall came amid market speculation that a third-quarter rally that lifted Brent to mid-2015 highs by late September had been overdone. A resumption in output at Libya's Sharara oilfield added to the concerns.

"Fundamentals may not yet be strong enough to support a continued rally, especially in growth-dependent commodities such as oil," Ole Hansen, head of commodity strategy at Denmark's Saxo Bank, said in a quarterly outlook to investors.

The Sharara oilfield restarted on Wednesday. It had been producing more than 230,000 barrels per day (bpd) before armed brigades closed it on Sunday.

Still, traders said a so-called market rebalancing is well underway, meaning demand is no longer undershooting available supply.

The rebalancing is a result of strong consumption and also efforts led by the Organization of the Petroleum Exporting Countries to cut output by around 1.8 million barrels per day (bpd) in 2017 and the first quarter of next year.

But rising oil production in the United States, which is not part of the output agreement, has prevented prices from climbing further.

U.S. output <C-OUT-T-EIA> hit 9.55 million bpd in late September, its highest level since July 2017, and drillers added six oil rigs in the week to Sept. 29, according to energy services firm Baker Hughes. <RIG-OL-USA-BHI>

"The number of active drilling rigs in the U.S. increased last week, highlighting the fact that higher oil prices will inevitably lead to more production from U.S. shale," William O'Loughlin, investment analyst at Australia's Rivkin Securities, wrote in a note to clients.

Data on Tuesday from the U.S. American Petroleum Institute showed gasoline stocks rising last week by a larger-than-expected 4.9 million barrels, with crude stocks dropping by 4.1 million barrels.

Traders said they would be watching for fuel inventory data from the U.S. Energy Information Administration, due on Wednesday, for further market guidance.

(Additional reporting by Henning Gloystein in Singapore; Editing by Dale Hudson)