I'm very happy to refer and vouch for someone that I've actually worked with (and look forward to working with again), but I'm not vouching for some random stranger...
In terms of reaching out, here are some things I did when I was job hunting:
1. The classic referral
Find the job post and work backwards from there (e.g., is there somebody I know (1st connection) or somebody who knows somebody I know (2nd connection) on LinkedIn who works at the company?).
If I knew the 1st connection, I'd reach out and ask if they were comfortable referring me.
2. The forwardable email
If it was a 2nd connection, I'd reach out with a forwardable email (https://also.roybahat.com/introductions-and-the-forward-intr...) and ask if they'd be able to forward an email and make an intro if they received a positive response.
3. Job hunting as an occasion
I made time to catch up with good friends. It felt energizing to get the moral support, with the added bonus that sometimes they knew people working at companies looking to hire. For example I would eventually get a job offer from Figma and that was because a good friend's partner worked there and was glad to refer me. I hadn't even heard of the opportunity before we talked.
4. The weak ties
I also made time to catch up with people I didn't know that well. There's some research on "weak ties" that suggest that people who you don't know well probably are exposed to a very different network to you, and will come across very different opportunities. The convo would be an opportunity for us to catch up and I'd talk about being open to job opportunities.
I hope this helps!
The proximity to my network doesn't need to be strong, but your resume does.
is how I've done it.
Worst case, you get ghosted. Best case, you get an interview.
Do you have experience with doing this in a way that doesn't come across as desperate or too forward? How close do you need to be with this network? Is Linkedin connection enough, or would you only do this approach with people you've worked with?