> How do you find such contract work? (at BigCo's I mean)

Become a half-human slave to one of large outstaffing companies with a proud yet faux label of "free contractor", for tax purposes.

Microsoft employs many contractors, we tend to call them "vendor staff" internally. More often than not these people will come from external companies who supply the staff, but they will be based on-site and embedded onto a team for the duration of their contract.

Pretty sure I can't name any of our vendor companies, but you can take a guess at some of the bigger ones I'm sure.

>How do you find such contract work?

It's a misnomer[0]. You usually find this work through vendor companies that actually hold the contracts with BigCo's. To the BigCo, you're a contractor but you're "technically" an FTE with the vendor company, who holds the actual contract (head count x duration) to work at the BigCo.

[0] - https://en.wikipedia.org/wiki/Misnomer

Knowing someone who works there and wants something done badly enough to go through the internal procurement process is one way.