The Federal Government allocated money to State Highway departments that would pay for ninety percent of the construction costs. They also demanded minimum standards be met and that the roads would only be built by Qualified Contractors. This meant they used union labor and were subjected to lots of audits, to prevent fraud and corruption. By 1960, when the Interstate Highway construction program shifted into high gear, there were 836 qualified contractors in the United States. They bid on constructing segments of the Interstate Highway system. These segments were usually fairly small, in the ten to twenty-five mile range. However, new contracts were awarded every year. One well regarded contractor, Cameron Joyce & Company, of Keokuk Iowa, ended up building most of the Interstates in Missouri and a large portion of the interstates in Illinois, Kansas and South Dakota. Ironically, they built only a small portion of the interstates in their home state of Iowa. These contracts were spread out over twenty years.
The Interstates that the Federal Government placed the highest priority on, the ones ending with a zero or five, which spanned the entire country, were built by multiple contractors. They were built in multiple locations, simultaneously. The workers did not follow the road as it was being built. On the other hand, these road construction jobs were well paid, so if a company like Cameron Joyce & Company won a contract to build a segment of the Interstate system in another state, they might keep their already trained workers and pay to relocate them closer to a new job site, rather than hire new workers in every town and county. To fully answer this question would require digging into the archives of the various State Highway departments, seeing which company won the contracts for what segments of the interstates that were built each year. That would be fairly easy, but to gain access to the payroll information of each of the qualified contractors would be more difficult. Many of the 836 Qualified contractors in 1960 have been merged together or have gone out of business when the Interstate Highway System was essentially completed in 1976. As a general rule of thumb though, I would lean towards the "new workers in every town and county" answer for the simple reason there were so many individual contractors that built the Interstate highway system.
Source: "The Roads that Built America: The Increadible Story of the US Interstate System" by Dan McNichol is a good overview.
For the finer details, the archives of the individual state highway departments keep tract of which construction companies built what segments of the interstate highways in their state.