You might have a personal marketing problem, or you might have an interviewing problem.
Small companies can be great. I tend to think "startup" when I hear small company, but I think the same rules may apply. Your tech stack is probably more fluid and you can learn new technologies more easily. These are great skills to have. But... if you are not clear about what type of work you want to do and how your skills apply to that work, it is easy to turn off a larger company because they can't fit you into a role (a "box"). A good recruiter will try to shop you around the company a bit, but there's a limit as to how much time they are willing to spend doing that.
Story: I recently had a set of interviews for a large company (the recruiting process was a mess and a joke) where I was bounced from place to place and nobody (myself included) could figure out where I fit in the company. I feel that company was very tight lipped or unsure about the roles they were hiring for, so I had no better answer than "I'd be happy in either position" to which one of the hiring managers said I would be a better fit for a startup. I came into the process very wishy-washy about what I wanted. Considering this didn't happen with any of the other companies I applied to, I blame this company. Everything was secretive and somewhat adversarial. I ended up at a much larger company with a much better reputation.
Point: If you are not clear about what you want, you may get stuck in the small company loop.
The good news is that you are getting phone interviews at larger companies. Before you take an interview, research the company and its culture, don't just rely on what you know about the company. Focus on what you have accomplished at your companies, and not the tech stack you use. I've heard a lot of candidates tell me about the complexities of the open-source stack they've worked with, and that's great, but I want to know how they will help us in our mission. Those types of nitty-gritty answers would better fit smaller companies, in my opinion (unless I am specifically asking about how they did something).
Also, larger companies have a pretty standard interview process. Data structures/algorithms and coding phone interviews. You need to practice, practice, practice on Leetcode or some other platform and even re-read an algorithms book. Smaller companies tend to be more holistic and use take-home projects, have you participate in a hackathon, whatever, but to get into a larger company, you need to learn how to play the interview game rather than just being a great coder (irony intended).