This doesn't seem to answer why an engineer is let go and gets rehired through an outsourcer.

In some cases it could be driven by the shape of the work & where the funding is allocated:

If there isn't enough guaranteed recurring work, it might not make sense to have a full time position, particularly in a country where its difficult to lay people off & if employees have additional overhead (pensions, employer funded heathcare or insurance, etc) vs contractors.

But, if there's funding allocated for some key project that's framed as a 6-12 month project, there might be a good business case to hire a contractor. Maybe the funding comes out of the project bucket, not the core funding for legacy product X bucket.

If the contractor is someone who was recently let go & has a good reputation within the company as someone who gets stuff done and is easy to work with, it's probably a no-brainer to re-engage them as a contractor vs rolling the dice on an unknown quantity.

Whoever is managing the budget of their old team gets a win as they were able to reduce headcount to fit in their budget

Whoever is managing the new project gets a win as they find a great contractor for their key project

The former employee returning as a contractor probably gets a win, as they get paid at a better daily rate while the project is rolling, provided they're able to line up more projects or land a new permie job once the project is completed.

If there's an outsourcer involved, they win by taking a cut. The former employee might also win by having the outsourcer involved if the company has some baroque process for engaging contractors with many compliance hoops to jump through -- in extreme cases (think banks, or public companies that need to demonstrate they don't do business with suppliers engaged in slavery, or so on) it could save the worker months of paperwork and tens of thousands in legal expenses to set up their own one-person agency and go through the compliance process to be able to work for their former employer, so they might not be able to win the contract work without piggybacking on an outsourcer who already has the contracts & compliance stuff sorted out.

most large companies have a 2-year limit on contractor employment so what they tend to do is they'll hire the same guy through a different contractor with another two-year agreement..... that's to get around the situation where if someone is working as a contractor for more than 2 years they can legally claim that they're actually an employee....see Vizcaino v. Microsoft Corp., 120 F.3d 1006 (9th Cir. 1997) [0]...

this is just a guess by the way but it seems like a plausible one, as I've seen it happen in Fortune 500 a lot, where the same guy comes back through a different vendor 2 years later if he was really good and they needed him to come back....

[0] https://law.justia.com/cases/federal/appellate-courts/F3/120...