KYC specifically refers to government ID requirements, not phone number verification. (But sadly many countries require government ID to buy even a prepaid sim card with cash.)

https://silent.link is good for KYC-free mobile data with global roaming, pay with Bitcoin. You would need a phone that supports e-sims.

https://jmp.chat is good for VOIP and SMS. It works with an XMPP client and accepts Bitcoin. However some websites may reject VOIP numbers for SMS verification.

https://smspva.com and https://juicysms.com/ are good options for anonymous SMS verification for Bitcoin. The numbers work in most cases, and if the number is rejected or you don't receive the SMS, you don't need to pay and can request a new number or try a different country. SMSPVA may re-use numbers, so make sure to set a password or PIN for your Signal or Telegram account (so that no one can login to it or re-register your number).

https://sms24.me, https://quackr.io and https://onlinesim.ru offer free SMS verification. These numbers are often already used for popular services (Gmail, Telegram, Discord, etc.) but may work for a less popular service.