UPDATE 1-Japan July-Sept steel output seen up 14% q/q-METI

* Record q/q output growth seen in July-Sept

* Gain driven by stock adjustment rebound, future unclear

* Output seen down 28.5 pct yr/yr (Adds details, quote)

TOKYO, June 29 (Reuters) - Japan's crude steel output will mark its biggest quarter-to-quarter jump on record in the July-September quarter, helped by public spending and a rundown in stocks, but the longer term outlook is still cloudy, the government said.

The Ministry of Economy and International Trade said on Monday that the country's crude steel output in July-September is expected to rise 14.3 percent from the preceding three months to 21.77 million tonnes.

That would be down 28.5 percent from a year earlier, however.

"We assume car production will jump nearly 20 percent in the next quarter from the current quarter, while government public spending will prop up demand from the construction and engineering industry," METI director Masaki Ishikawa told a news conference.

He also said demand for exports could rise 20 percent thanks to progress in inventory adjustments at steel users in China, Thailand and other Asian countries.

But he added strong growth in output may not be repeated in October-December because stock adjustments will likely be completed by the end of the next quarter.

"It is still very difficult to foresee the trend in real demand after October," he said.

METI forecast July-September demand for steel from Japanese industry at 17.57 million tonnes, up 10.2 percent from April-June but down 20.4 percent from a year earlier. (Reporting by Yuko Inoue; Editing by Joseph Radford)