You would need to add 'to' to your sentence eg. After assembly, the cars go to the paint shop at the factory....
Not knowing much about car production, I know what you are trying to say, however I cannot think of an alternative. I have heard 'spray booth' being used when talking about car painting though.
I think you could say, 'After assembly, the cars go to get painted at the factory....' Or, 'After assembly, the cars then go to get painted within the factory...'
Hope this helps.
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