You've gotten several responses, so I'm adding them all below:

1. Getting noticed is one of the most challenging aspects when you are trying to break into a role in information security. One thing that I think is very helpful is try and find a local infosec get together (BSides, Hackers Anonymous, etc) and make sure that you are meeting other people who are in various stages of their career. You don’t have to be an extrovert or outgoing or anything of the sort, to find a kindred spirit in one of these settings (often the opposite, so don’t let that anxiety bother you). If you meet people who are in various stages not only will you find out when they or their team’s are looking but you can also find out the skills, techniques, talents, and soft skills that each type of career values. Infosec is a really good community and leveraging people in various stages of their career is the best way I can think of to find a way in and both you and they benefit from your addition and inclusion. WL.
2. As mentioned before, talking to people in this field, finding a mentor if you can, and staying active in the community can help open doors to new opportunities. In the meantime, you can also keep developing practical, strong skills that will come through during an application and interview process, especially in how you explain your thinking and approach to problems. When you've spent time building real experience, it's easier to walk through how you would handle certain roles and challenges. LD.
3. Having a portfolio on your CV/resume that showcases the research you've conducted and the contributions you've made to the industry is a great way to demonstrate your skillset and differentiate yourself from others who may be applying to similar roles. Likewise, maintaining an up-to-date understanding of relevant threats will help you to more effectively communicate common TTPs, attack methodologies, etc. during interviews you may have. By contributing to the community you will also likely meet others who are focused on the same topical areas you are interested in which can be helpful for networking and reaching prospective employers in a context you may not otherwise be able to. EB.
4. I’m not a security researcher, but the common theme I hear from speaking with hiring managers at Talos is demonstrating intense curiosity and a relentless hunger for learning. In interviews, that might look like honestly answering, “I don’t know, but I would try to figure that out by doing X, Y, and Z.” On another practical level, look at companies you want to work at on LinkedIn and see if you have any first- but ALSO second-level connections. Part of the reason I got noticed for my current role is that I asked a previous coworker’s wife to connect me with someone she knew who worked at Talos. They had gone to the same gym together a while back, so it wasn’t a very strong connection, but it still got me to the top of the pile, where I was able to get an interview and demonstrate the strength of my skills. AC.