(Adds recent COVID cases, Nomura estimate on cities under lockdowns)

BEIJING, July 5 (Reuters) - China's services activity snapped three months of decline in June and grew at the fastest rate in almost a year as easing COVID curbs revived demand, although firms remained cautious about hiring, a private-sector survey showed on Tuesday.

The Caixin services purchasing managers' index (PM) rose to 54.5 in June, indicating the fastest growth since July last year and the first expansion since February. That compared with 41.4 in May.

The 50-point mark separates growth from contraction on a monthly basis.

The upbeat findings are in line with those in an official survey that showed services industry staged the fastest rebound in 13 months, with sectors that were hard hit by COVID curbs such as retail and road transport catching up with previously depressed demand.

While activity has sped up since various COVID lockdowns were rolled back in June, headwinds facing the world's second-largest economy persist, including a still subdued property market, soft consumer spending and fear of any recurring waves of infections.

Cases have risen in the past few days. Daily infections in mainland China increased to more than 300 over the weekend compared with a few dozens in late June.

On Monday, 11 cities were in full or partial lockdowns or had implemented district-based controls, compared with five cities in the previous week, according to Nomura.

The Caixin survey on Tuesday showed a sub-index for new business jumped to 52.4 - the highest this year - from 44.8 in the previous month while the drop in export orders also softened. Price pressures also eased, with input prices largely unchanged from the previous month.

However, employment continued to decline in June for the sixth straight month, with services firms linking it to cost-cutting initiatives and resignations amid COVID.

"Overall, regional COVID outbreaks were put under control and restrictions were loosened in June, facilitating a gradual recovery in business operations," said Wang Zhe, Senior Economist at Caixin Insight Group.

"Deteriorating household income and expectations caused by a weak labour market dampened the demand recovery. Correspondingly, supportive policies should target employees, gig workers and low-income groups impacted by the outbreaks."

Analysts expect further improvement in conditions in the third quarter, although the official GDP target of around 5.5% for this year will be hard to achieve unless the government abandons the zero-COVID strategy.