By Kevin Buckland

TOKYO, June 1 (Reuters) -Most Asia-Pacific stock markets rose on Thursday amid receding bets for a U.S. rate hike this month and relief over the passage of the U.S. debt ceiling bill through the House.

The dollar sagged to a one-week low versus the yen and hung close to Wednesday's more-than-two-month trough to the euro after Federal Reserve officials including Governor and vice chair nominee Philip Jefferson pointed to a rate hike "skip" at the June 13-14 policy meeting.

Treasury yields rose slightly from nearly two-week lows.

Crude oil prices edged off four-week lows following a surprise swing back to growth in a private survey of Chinese factory activity, with an OPEC+ meeting looming on the weekend.

MSCI's broadest index of Asia-Pacific shares .MIAP00000PUS gained 0.45%, rebounding after touching the lowest level since March 22 in the previous session.

Japan's Nikkei .N225 added 0.29%, while Hong Kong's Hang Seng .HSI gained 0.5% and mainland Chinese blue chips .CSI300 advanced 0.53%.

A divided House passed a bill to suspend the $31.4 trillion debt ceiling - and avert a catastrophic default - with majority support from both Democrats and Republicans, stoking optimism that it can move through the Senate before the weekend.

"This has gone through with a very big majority, so there's enough bipartisan support that it's very hard to believe this isn't going to be even more of a formality in the Senate," said Ray Attrill, head of foreign-exchange strategy at National Australia Bank.

""What this does is it turns the attention to the incoming data and the Fed meeting this month," Attrill added.

Money markets currently lay about 38% odds for a hike on June 14, swinging back from about 70% earlier in the day, after some unexpectedly hot jobs numbers. FEDWATCH

However, shortly after, the Fed's Jefferson said skipping a rate hike in two weeks would provide policymakers time to see more data before making a decision. Philadelphia Fed President Patrick Harker also said on Wednesday that for now he is inclined to support a "skip" in rate hikes.

The dollar was little changed at 139.435 yen JPY=EBS after slipping to the lowest since May 25 at 138.96 earlier in the session.

The euro EUR=EBS was flat at $1.06905. It sank as low as $1.0635 on Wednesday for the first time since March 20.

Benchmark 10-year U.S. Treasury yields US10YT=RR edged up to 3.6655% in Tokyo, after dipping to 3.6140% overnight for the first time since May 18.

Brent crude futures for August delivery LCOc1 rose 46 cents, or 0.63% to $73.06 a barrel, while U.S. West Texas Intermediate crude (WTI) CLc1 added 40 cents, or 0.59%, to $68.49 a barrel.

Reporting by Kevin Buckland; Editing by Simon Cameron-Moore

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