I am in the same boat as you. For generalist,after a point, i feel the only way up is to start something of your own.
Just give it some time and start by doing it on a part-time basis. It might take several months before you land your first project, but once you completed some work for some clients, more business will come if you are good.
You will know when you are ready to take your individual contracts full-time.
You can just drop it in casual conversation. "What do you do for a living?" "Oh I'm a software developer at X-Corp, and I also do some work for individual clients" Something like that. People remember that stuff. If you like, you can even tell people you offer a finder's fee. Just don't be pushy. Be willing to build the business part-time until you have enough work to take it full-time.
You probably won't be getting big projects from the get go, but if you are humble, that won't be a problem. My first project was just optimizing and cleaning a slow and messy Python script. I charged $35 an hour an threw 10 hours into it.