(Refiles to change release date of story in first paragraph)

BEIJING, Nov 3 (Reuters) - Activity picked up slightly in China's services sector in October but growth remained modest and much weaker than historical trends, a private survey showed on Friday.

The findings, along with other private and official business readings earlier in the week, are likely to reinforce views that the world's second-largest economy saw a more subdued start to the fourth quarter after racing ahead earlier in the year.

The Caixin/Markit services purchasing managers' index (PMI) rose to 51.2 in October, up slightly from September's 21-month low but suggesting only a subdued expansion in activity and new orders.

A reading above 50 indicates growth, and any lower signals deterioration on a monthly basis. The index had hit a three-month high in August.

An official gauge of the non-manufacturing sector on Tuesday showed slower but still relatively solid growth in October.

China's leaders are counting on growth in services and consumption to rebalance their economic growth model from its heavy reliance on investment and exports. The services sector accounts for over half of the economy, with rising wages giving Chinese consumers more spending power that is being felt at home and abroad.

Unlike the official data, the Caixin survey tends to focus more on small and mid-sized companies, which have tended to be under greater strain than their larger, state-supported peers.

But one positive takeaway from the latest Caixin survey suggested smaller firms may be regaining some pricing power.

While their input costs for raw material, fuel and labour continued to rise, service providers were able to raise their prices at the quickest pace in over two years, protecting profit margins but also pointing to building inflationary pressures.

China's economy has surprised global financial markets and investors with robust growth of nearly 7 percent so far this year, driven by a renaissance in long-ailing "smokestack" industries such as steel.

But property and construction activity, two of the economy's main growth drivers, are starting to cool under the weight of government measures to cool heated housing prices and higher borrowing costs.

While growth in services took up much of that slack in the third quarter, manufacturing may face a sharper slowdown in coming months as the government orders many mills and factories in northeast China to curtail or halt production to reduce winter air pollution.

Reflecting the growing drag on manufacturing output, Caixin's composite manufacturing and services PMI for October fell 0.4 points to 51.0 in October, the lowest since June 2016.