* International Energy Agency sees slower oil demand growth
* Crude futures head for first weekly drop in five weeks
* Hurricane causes 40 pct drop in Gulf of Mexico oil output
* U.S. oil rig count rises for first time in month -Baker Hughes (Adds settlement prices, CFTC data, Gulf of Mexico output)
NEW YORK, Oct 12 (Reuters) - Crude futures steadied late in the session on Friday, following the stock market slightly higher after earlier swinging lower on a weakening oil demand outlook.
Stock markets worldwide bounced back on Friday after a multi-day sell-off but remained on track for their biggest weekly losses in months, while U.S. Treasury yields inched higher and the dollar held its gains.
The move prompted oil futures to tick slightly higher late in the day after coming under pressure from a bearish demand forecast.
The International Energy Agency, the West's energy watchdog said in its monthly report that the market looked "adequately supplied for now" and trimmed its forecasts for world oil demand growth this year and next.
"This is due to a weaker economic outlook, trade concerns, higher oil prices and a revision to Chinese data," said the IEA, which advises industrialized countries on energy policy.
Brent crude settled up 17 cents a barrel at $80.43, after dropping 3.4 percent on Thursday. U.S. crude futures rose 37 cents to $71.34 a barrel.
"The weaker outlook has gotten a raised profile in the market, but there's potential for a real supply crunch toward the end of this year," said John Kilduff, a partner at Again Capital Management in New York. "The demand outlook is hurt right now because of the situation with the U.S. and China in particular."
Both benchmarks fell for the first time in five weeks, pressured by a big rise in U.S. inventories and fading concerns about shrinking global supplies due to looming U.S. sanctions on Iran's oil exports. U.S. crude was down 3.6 percent on the week, while Brent crude fell 4.1 percent.
The IEA report is the latest official forecaster to predict weaker demand ahead and conclude that supply is adequate. The Organization of the Petroleum Exporting Countries (OPEC) made a similar move on Thursday.
"The bearish alarm bells are ringing for next year's oil balance as market players brace for the return of a supply surplus," said Stephen Brennock of oil broker PVM.
Additionally, the U.S. oil drilling rig count rose this week for the first time in four weeks, an indication that production is rising.
Drillers added eight oil rigs in the week to Oct. 12, bringing the total count to 869, General Electric Co's Baker Hughes energy services firm said in its closely followed report on Friday. <RIG-OL-USA-BHI>
The increase is the biggest weekly gain since mid-August.
Money managers cut their net long U.S. crude futures and options positions in the week to Oct. 9, the U.S. Commodity Futures Trading Commission (CFTC) said on Friday. The speculator group cut its combined futures and options position in New York and London by 36,652 contracts to 296,456 during the period.
A drop in U.S. oil production this week supported prices. U.S. Gulf of Mexico producers have cut oil output by 32 percent and natural gas production by 13 percent as a result of the lingering effects of Hurricane Michael, the Bureau of Safety and Environmental Enforcement (BSEE) said on Friday, citing reports from 27 companies.
The reductions continued as oil and gas companies moved more workers back to production platforms that were evacuated earlier in the week. As of Friday morning, nine platforms were still unoccupied, BSEE said in a daily update, down from 89 platforms on Wednesday.
(Additional reporting by Aaron Sheldrick and Alex Lawler; editing by Rosalba O'Brien and Diane Craft)