What's the best way to go about finding a job through LinkedIn?
Should I just start sending invites to connect to startup founders and then when they accept, make my case about wanting to work for them?
While I am listed on job sites and apply for available positions on there, I'd like to somehow do outreach directly to startups.
I'd like to answer the question, "What's the best way to go about finding a job?" in general. LinkedIn is just one channel out of many others, though it can help a lot.
1) First, have a target industry or set of companies. Don't worry about how narrow this may be at first, you can gradually widen it as you exhaust your list. But it can help to have some targets in mind first.
2) Second, look through the job listings of those companies. This will give you an idea of what kinds of roles they offer, and the kinds of people they hire. It is always possible that a company may be willing to create a new role for you if they like you enough.
3) Third, find out if you have any connections to people working at those companies. This is where a service like Hachi can really help. This service looks at your LinkedIn, Facebook, and Twitter networks to find a path of connections to specific people. And you can search by a specific person's name, as well as a company in general.
4) Fourth, reach out to those people via your connections. Don't expect all of your connections to help out. Some people don't like referring others, and other people may not know the target person well enough to speak on your behalf. Be polite, respectful, and considerate (I shouldn't have to write this, but some people surprisingly don't do this and really need to hear it.)
5) Fifth, have a personal pitch ready. This can be a cover letter, a brief email intro, etc. This basically needs to tell the target person what you want, and why you would be good for that role. I wouldn't really encourage reusing the same message with everyone; a personal touch can go a long way. For extra bonus points, create a mini-website explaining why, in a creative way, you are great for this role. I've seen some people do this with amazing results.
If you are lucky, doing all of that should at least land you a phone call. Good luck!