This is the most straightforward and best method of hiding presence from your contacts.

1. Get a new SIM card or choose a secondary phone number to use for Telegram.
   
2. In Telegram, go to Settings > select your phone number > Change Number.

If you're new to Telegram and want to start safely, simply register with your secondary phone number.

This method can be useful, but you need to be careful.

1. Reject contact access permission for your Telegram client in system settings.
   
2. Under Settings > Privacy and Security disable Sync Contacts and select Delete Synced Contacts.
   
   • Some versions have additionally buried this under Privacy and Security > Data Settings (thanks Kijame)
3. Select the Finding by Phone Number setting (may be under Phone Number > Who can find me by my number?) and set it to My Contacts. (Even though the setting suggests that your contacts will be able to find your account, if none are synced, none apply.)

Important!

Whenever you reinstall your Telegram client, make sure not to allow contact access and to repeat the first two steps above! Telegram tends to automatically switch on contact sync if you re-login.

Also note, you may set your Profile Photo visibility to Everybody since your contacts can no longer view your profile in the first place.