SAN FRANCISCO, June 24 (Reuters) - The Federal Reserve must "exhibit care" as it aims to finish the job of bringing inflation under control, San Francisco Fed President Mary Daly said on Monday, noting that rising unemployment is increasingly a risk.

"We must continue the work of fully restoring price stability without a painful disruption to the economy," Daly said in remarks prepared for delivery to the Commonwealth Club in San Francisco. And while there is still "more work to do" on bringing inflation down, she said, "inflation is not the only risk we face."

Reducing inflation further will likely require restraining demand, she said, and while so far the unemployment rate - now 4% - remains below long-run sustainable levels, "future labor market slowing could translate into higher unemployment." To avoid that, she said, the Fed must be both "vigilant and open."

The Fed earlier this month left interest rates in the 5.25%-5.5% range where they have been since last July, and policymakers signaled they may cut interest rates only once this year, rather than the three rate cuts they expected back in March.

Daly did not say how many rate cuts, if any, she believes the Fed will need to deliver to navigate between the twin risks of still-too-high inflation and the potential for rising unemployment

As this point, she said, policy must be "conditional" and ready for any of several possible paths, including holding rates where they are for longer should inflation disappoint, or cutting them should the labor market soften more than expected.

And, she said, "if we continue to see gradual declines in inflation and a slow rebalancing in the labor market, then we can normalize policy over time, as many expect."

Inflation by the Fed's targeted measure, the year-over-year change in the personal consumption expenditures price index, was 2.7% in both April and March, up from readings of 2.5% in both February and January.

More recent data suggests the index may not have risen at all from April to May, according to economists' forecasts.

"The bumpiness of inflation data so far this year has not inspired confidence," Daly said. "Recent readings are more encouraging, but it is hard to know if we are truly on track to sustainable price stability." (Reporting by Ann Saphir; Editing by Andrea Ricci)