(Updates as of 1428 ET)

By Alden Bentley and Karen Brettell

NEW YORK, May 9 (Reuters) -

Treasury yields slipped on Thursday amid relief that all

$125 billion in new note and bond supply this week was absorbed

smoothly, allowing traders to focus on pivotal inflation reports

for the Federal Reserve's higher-for-longer rate strategy.

The Treasury sold $25 billion in 30-year bonds on Thursday,

at a high yield of 4.635%, lower than where the yield on the

screens was at the close of competitive bidding, indicating

strong demand.

A bid-to-cover ratio of 2.41 was higher than the market

was expecting, according to Subadra Rajappa, head of U.S. rates

strategy at Societe Generale, New York.

The 30-year yield fell after the auction,

showing the result was well received, and was last down 1.5

basis points from late Wednesday at 4.6166%.

"We actually had a pretty decent 30-year auction, all

things considered," Rajappa said. "This whole week has been

pretty much about Treasury supply and corporate supply. It's a

wait-and-see game until CPI next week."

The government got solid interest at a $42 billion

auction of 10-year notes on Wednesday and a $58 billion sale of

three-year notes on Tuesday.

The April Producer Price Index report comes on Tuesday,

and the closely followed Consumer Price Index next Wednesday.

Combined they will provide insight on whether inflation has

resumed its downward trend toward the Fed's 2% target rate.

"There is a reason the CPI has supplanted the employment

report as the biggest market mover on the economic calendar,"

said Chris Low, chief economist at FHN Financial in a daily

client note. "As we try to anticipate how the Fed conversation

will evolve next, it will depend more than anything on what CPI

inflation does next Wednesday."

In early trade, benchmark yields briefly dipped after

news that Initial claims for state unemployment benefits

increased 22,000 to 231,000 last week. That was higher than the

215,000 expected by economists polled by Reuters and could be

good news for the Fed as further evidence that inflationary

labor-market tightness is ebbing.

"If you just read it on the surface, it looks like one of

the uglier numbers that we've seen in the last several months,"

said Thomas Simons, a money market economist at Jefferies in New

York, but "volatility around the first of the month is not

unusual."

Yields fell hard on Friday after payrolls for April came

in below expectations. That followed the Federal Open Market

Committee meeting, where it held rates steady but said it still

expects a rate cut to be its next move even as inflation remains

stubbornly high.

Traders are pricing in the probability of two 25 basis point

cuts this year, with the first expected in September, but any

cuts will likely depend on whether inflation can resume its

easing trend.

The benchmark 10-year note yield was last off

2.6 basis points at 4.457%. On Tuesday it hit 4.42%, the lowest

since April 10

Two-year yields, which typically move in step

with interest rate expectations, fell 3.2 basis points to

4.8112%, remaining in a range since Friday's fall to 4.806%,

their lowest since April 5.

The inversion in the yield curve between two-year and

10-year yields deepened more than a basis point

to minus 35.6 basis points.