How can I add new phone number to activate free Apple TV?

To edit phone numbers associated with your Apple Account, do so on your iPhone in Settings, or else on the web:

Settings﹥[You/Your Apple Account]﹥Sign In & Security﹥ᴇᴍᴀɪʟ & ᴘʜᴏɴᴇ ɴᴜᴍʙᴇʀs﹥…

• Also remove old phone numbers that may be in use for two-factor authentication (Settings﹥[You/Your Apple Account]﹥Sign In & Security﹥[sᴇᴄᴜʀɪᴛʏ] Two-Factor Authentication﹥Add or remove trusted phone numbers).

Manage your Apple Account on the web - Apple.

• Also remove old phone numbers that may be in use for two-factor authentication (﹥Sign In & Security﹥Account Security﹥Add or remove trusted phone numbers).

To use the offer from EE, do as described on the offer page for Apple TV+ through EE. At some point in that process, EE will ask for your Apple Account email, to link this subscription to. Your phone number isn’t relevant to Apple to get this subscription in this way. The link between this offer subscription and your EE phone number exists only in EE’s administration.

if you mean that you wish to use a new appleID to be able to use free X time of appletv+ more than once then you maybe have to read

How to create a new Apple Account - Apple Support

if it's the same phone you wish to use then you should first take the following steps to prepare for the new appleID

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support