Way back in the last economic downturn, the dotcom/bomb era, job-seekers often had to choose whether to focus on a full-time job search or take consulting projects. Lots of employers were wary of hiring people who might take a full-time job for port-in-a-storm reasons and then bail again to return to consulting when the economy improved. Today, I see none of that sentiment around.
The world has changed enormously. I encourage every job-seeker to hang out a consulting shingle. Nothing bad and lots of great things can come of it.
For one thing, consultants and other entrepreneurs understand better than most working people how they make a difference every day. If your work wasn't good, you wouldn't have any clients. So for hiring managers, you're more credible, not less credible, when you have a consulting track record and current clients.
Doing consulting will help keep you sharp. It's very different solving one defined problem for a client than showing up and doing what you're told every day as many full-time jobs require.
Consulting will teach you how to bid jobs, how to set your prices, how to define projects and how to manage them. It's invaluable experience, and I recommend it for everyone remotely involved in business.
Consulting will give you income, a place to go, new contacts and self-esteem that can only help your job search. You might build so much business that you put off the job hunt and focus on your practice.
A consulting gig could morph into a full-time job. There are no downsides; pretty much every client assumes that any consultant could accept a job and fall off the consulting scene at any time, just as anyone can change directions at any moment. You can simply say to clients "I'm consulting and looking at some full-time opportunities, too."
You don't need lots of preparation to begin consulting. I encourage you to chat with your accountant and your lawyer about whether you need a corporate entity. You'll need a business card, and those are easy to get for a few bucks at an office.
You don't need a fancy business name. You could put your own name on the business cards, along with a very short description of what you do (or none at all if you prefer to be inscrutable) and your contact info (e-mail, phone and LinkedIn profile address).