I can think of a few ways to solve this problem but there's only one I really like. And that is because the truth of the solution is not out there in the marketplace, but here inside the mind of the contractor.

What does this freelancer believe is "a lot of money"?

That's your first and key question. It drives all the others. You have to be totally honest about this because people often lie to themselves about this number...the reality is the truth is a figure a whole lot lower than they initially say.

Why are we doing this? Because if you mismatch your beliefs with your target market, you will never sell. Either the projects will be too small for your taste, in which case you'll find ways to screw up the sale because the ideas are boring...or (more commonly) the projects will be valued too highly for your personal sense of value, and you'll screw up the sale because it's too big for you.

The sweet spot is a level that is just a small bit higher than what you think of as "a lot of money." Let's say you get real with yourself, you've never earned more than $50K a year (most people have not), and so you're used to a salary of around $4K. Earning $10K or $15K at once is frightening to you. Exciting, but scary. You don't really believe you can do it...that someone will actually pay you that in one go.

So looking for companies that have a $10-15K or higher project level comfort zone is a bad idea for you at present in this example. Projects at $5-7K are a much better choice, because they will draw you on while seeming believable in terms of someone really sending you that much at once. And if you're used to twice monthly salary payments, $3-5K may be even better.

Once you understand what is true for you right now--and the goalposts can be changed; this is all habit and you have to be consciously aware of it and work on it continuously--then you can go out and look for a match in the marketplace.

The fact is there are customers at every level. I had a South African friend I worked for about 10 years ago who had a design agency there before he moved to Vancouver, Canada. One of his designers bid on a job to redesign a credit card cover, and that art project was awarded to them at $80K.

$80K to redesign that small space.

Why? Because that customer felt it was worth it. But if the designer hadn't believed THEY were worth it, they would never have bid on that project, not at that amount...heck, they may not have even SEEN the listing for the project because their RAS would have screened it out.

So find your sweet spot, then go and find out who has projects that are worth 10-20X that figure. It's easy to get someone to say "Yes" to a project where they know the value is $100K when your price is $5K to solve the problem. Maybe even $10K as the investment to fix things. And $100K problems are lying around everywhere in business.

As for what niche, I like to pick niches I enjoy talking about all day. Beyond typical project value that's another good factor. If I'm going to be stuck in there, I might as well enjoy it. Also makes the marketing easier, since I stand out due to knowing the industry jargon. Yeah, a designer could work on anything...but that's the wrong perspective to be using when it comes to finding clients. The client's perspective is this:
"Is this for me? Will this work for me?" And appearing general does not help answer those questions confidently.