OK, now that we're several weeks into the update, the drop rate is a bit clearer based on observations.
Contracts do NOT use the Gun Mettle drop times.
What is the same between the contracts are this:
- You start with 2 contracts.
- You get 2 additional contracts per week (including the first week).
However, when you get those contracts has changed drastically.
Based on available evidence, it appears that contracts drop on two fixed days specific to your account.
Contracts for my account drop on Thursday mid-day and Friday early morning. At first, I thought this was based on the Item Drop system (which resets at midnight UTC Thursday morning), but others have commented about how they got contracts sooner or later.
This is also why contract #4 seemed to take so long to appear for me: The update hit on a Thursday afternoon, so it took roughly 6 days and 18 hours before I got contract #4.
As I noted in my (now deleted) answer, Valve plans to have exactly 26 contracts total.