The April figure was 16% lower than the same month last year, and a quarter lower than March, when numbers were likely to have been boosted by manufacturers shipping more cars to the US before President Trump's 25% tariff on steel, aluminium, and cars kicked in.
On Wednesday, a US court blocked many of Trump's tariffs, but the ruling does not apply to the tariff on steel, aluminium, and cars.
British car maker Jaguar Land Rover (JLR) is paying 27.5% tariffs on everything it ships to the US, which it said is costing it "a huge amount of money".
The firm sends cars from its UK business to its US business - meaning it pays both export and import taxes on any cars it sends across the Atlantic.
The company also said it is frustrated that the new deal agreed in early May between the UK and US to reduce tariffs on cars to 10% up to a quota of 100,000 vehicles is taking so long to come into effect.
The fact that Easter fell in April this year, which meant there were fewer working days, was another reason car making fell, the SMMT said.
The lowest April output before that - outside the pandemic - was back in 1952, when 53,517 vehicles were produced.
Car production for exports fells by 10.1%, said the SMMT, driven by falls in demand from the UK's biggest export markets the US and EU.
The group said the total number of vehicles manufactured in the UK for the first four months of the year was the lowest since 2009.