I've been a fractional for over a year now.
I went in with one conviction:
Being a founding designer at 0 to 1 startups is messy. Product design (not just UI) is iterative, and as an early team you're constantly changing direction.
So the best way to deliver value is being embedded. You learn the space, influence what they build/ship and help define the design role. This compounds over time.
Turns out I was right.
Call it fractional, consulting, contracting etc. The point is you're embedded part-time over a mid/long term engagement.
It's not for everyone, but this way of working is exactly what works for me.
Honestly it depends on the kind of startup.
Some, like funded startups are actively looking for long term design help. They want to grow fast and build a team, and have the budget to keep you around. So you can propose long term directly.
Others like bootstrap or lean teams aren't ready to commit upfront. That's fine. Start with a small scope, show value, let it grow organically.
Either way, network and referrals make it easier at the start because trust is already there.