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Is it a dumb idea to work at a firm for a year after graduating law school then do a state Supreme Court clerkship followed by a federal COA clerkship?

No, that would be a good idea, especially if it is a well-regarded SSC.

I think some nuance to the answer above is helpful. If you already have a federal CoA clerkship lined up and are considering a SSC, I think the situations where it’s worth it are more limited. Unless you plan to practice in the state, I don’t think the SSC would be worth it over a district court clerkship or even an extra year of biglaw pay.

I don't think your question clearly presents the issue you want answered. Is the whole idea bad? No. People do it all the time. Are you asking if it's a bad idea to fill a gap between your COA with a SSC rather than work in biglaw for another year? If so, the previous anon is correct, it's based on your state. Are you asking if it's a bad idea to follow a SSC with a COA? If so, definitely a great idea, and better to knock it out now than going back to practice before doing so later down the road.