Send them a postcard or a letter with your phone number written clearly at the top. That is the fastest and most reliable method. Mail reaches inmates even when phone access is still being set up, and once they have your number in hand they can add it to their approved call list and reach out.
InmateAid's postcard and letter service lets you send from your phone or computer from anywhere in the country, and delivery through the US Postal Service typically takes a few days to reach the facility. Your home address does not appear on the envelope, which keeps your personal information private.
One letter with your number is all it takes. Keep it brief and direct so the number is impossible to miss.