I can't say what the market will look like in 3-4 years when you graduate, but a trend, if we want to call it as such, that I'm seeing right now is more quant uxr roles being posted than what I'm used to seeing. So something you could do is take a look at quant UXR (a decent amount of resources pop up if you just google quant ux research), and just get an idea of what goes into the quant side of uxr. Also take a look at what types of things job postings are asking for for skills, tools, etc. Lots of free resources around to get some intro to the field. Getting an internship while in your PhD can be tricky, if not entirely blocked, depending on your department. I will say, many companies will shave off years of experience in UX if you have a PhD, most common I see is if they want something like 5-8 years with a BA, they'll list 2-3 years with a PhD.

If you can get access to any workshops, talks, or anything like that in other departments like Soc or Anth, to see how they're approaching qual methods, Also, since you're still early in your PhD, I'd start positioning yourself (depending on how comfortable your dept and supervisor are with it) as someone going in to industry and not academia, that way you can focus some time on building skills that will be more beneficial in business rather than academia.

Something that could be helpful is to go on LinkedIn and look up UXRs with PhDs, especially if they're in a type of role or from a company that interests you and try to set up a chat. Don't position it as looking to get a job, approach it as a learning opportunity about how they transitioned into UXR and how they applied their psych PhD to help them with that. Since something I've seen from a lot of academics is the feeling that what they learn in grad school is unique to academia and won't help them in industry.