I think so, yes. If you read the actual rule (https://www.ftc.gov/system/files/ftc_gov/pdf/noncompete-rule.pdf) it says a noncompete is something against “seeking or accepting work in the United States with a different person”. Since you’re working in the first part with a contracting agency, and then going to work with a different company, the rule seems applicable here.

I’m not sure why they use “person”, but I’m assuming your W-2 or 1099 would have different companies, and the different companies would have different presidents/CEOs/chairmen, so it would objectively be different both in the general legal Romney-style “corporations are people” person and the literal dictionary person.