You have a few options, and which one works fastest depends on your situation.
The most reliable method is sending a letter or postcard with the new number written clearly in the body of the message. InmateAid's letter and postcard service works well for exactly this kind of thing, and your personal address stays private in the process. Allow the usual 6 to 7 days for it to clear the mailroom and reach your inmate.
If timing is tight and your old number is still active, have your inmate call it one more time so you can relay the new number verbally during that call. That is often the quickest bridge while the letter is in transit.
The third option is going through facility administration and asking them to pass the number along. It does happen occasionally, but it depends entirely on the staff and the facility. Do not count on it, but it is worth a try if you have a contact there who is cooperative.
The letter is your safest bet. Get it in the mail as soon as possible so there is no gap in communication once the old number goes inactive.