What is the best marketing method to attain consistent work and high-paying clients as a freelance designer (UI & UX)?

If you consider yourself one of the best (which I assume you do, given that you're looking for high-paying clients), you should look into TopTal https://www.toptal.com/talent/apply

TopTal is a freelance site similar to UpWork, Freelancer, Odesk, etc...the difference is, TopTal ONLY hires the top 3% (so bring your A++ Game) and they only allow the top companies to hire their freelancers, so you don't have to worry about competing lowball bids to get work. Supply is low, demand is high, and you could literally pick up full time work immediately after being approved. The application process is pretty intense but if you get through it it's worth it. They are rabidly looking for UI & UX designers.

If you're not already using Upwork, I would get on there too. Some of the companies on there are looking for quality over price and willing to pay more for good work, but the demand is much lower.

I help small companies and startups market their services.

First step is to define your target market. All big companies will have PR firms as their retainers. So talk to PR firms.

Host some of your work/ templates on your web site as samples. You can even sell them from there.

Start talking more about UX and write articles and case studies about the same. Provide free advise in these articles and tips for doing the same. Remember that these tips are useless unless I am skilled enough to implement so you are not giving away any IP.

Big companies wont hire freelancers usually. So figure out who is the channel and work through them.

Please feel free to call me if you wish to discuss the same.
Best regards,
Sridhar

1 of the projects I haven't yet touched would be developing UXDesigners.com, which I own. Ditto AppDesigners.com.

My business model would be based around helping UX designers acquire customers.

So far, I don't know much about the challenges UX designers like you are facing. That means I have no advice for you – none that I'm confident enough in to charge for, anyway. But I am interested in your problems, in case my project can one day solve them.

We can set up a free call, if you'd be interested in sharing your perspective.