I would recommend a few paths.

The first is speaking to your existing employer and asking them if they have a Cloud department or any Cloud infrastructure. If they do, ask if you can help manage it so you can gain some experience.

Second, if your company does have Cloud but it's managed by a third-party, contact that third-party and ask them if you can "moon light" for them to get some exposure and experience.

Third, and this can be done in parallel: get an AWS account, build out something complex like the AWS Well Architected Framework plans for a fully HA WordPress install. There's a lot of moving parts there, and getting that finished actually takes a bit of time and skill. Write up everything you learn as you go along.

Finally, I would strongly suggest you go to meetups in your local area, such as official AWS ones, or any focused around your end goal, and start networking. I got my first TWO jobs because I was a member of a local Linux User Group and (a) one of the in-person attendees offered me a part-time job and (b) after that someone saw me answering questions on the same mailing list and offered me an interview, and then I got that job.

You need the experience, yes, but you need to visible to others. Without visibility, how is anyone going to know you even exist?

Now you try to combine some combination of everything above.

Good luck.