But the industry body warns future growth may be hit if the government does not maintain unrestricted access to markets in the European Union (EU).
The UK made 158,641 cars last month, 10.4% more than in June 2015.
The SMMT says the growth was driven by decisions to invest on the basis of demand from its biggest market, the EU.
"These decisions were based on many factors but, primarily, on tariff-free access to the single market, economic stability and record levels of productivity from a highly skilled workforce," said Mike Hawes, SMMT chief executive.
There are signs from elsewhere that the auto sector in the UK is already being affected by the UK's vote to leave the EU.
Car dealer Inchcape has warned that the Brexit vote is expected to slow growth in UK new car registrations in the coming months.
"Ahead of the EU referendum, the second quarter new vehicle market growth rate moderated to 1% from 5.1% in the first quarter," it said. "We expect this moderation of the new vehicle market to persist into the second half of 2016."
Second quarter GDP data, released on Wednesday, was broadly positive but also highlighted the potential for a slowdown in manufacturing for the car industry and other manufacturing.
In April industrial production jumped 2.1% month-on-month, but then dropped 0.5% in May and rose only 0.2% in June.
UK car production jumped more than 10% year-on-year in June, says the Society of Motor Manufacturers and Traders (SMMT).