You can add the private key by following the below steps:

  1. Open VSCode
  2. Press F1 and search for "Remote-SSH: Open Configuration File"
  3. Select the config file to edit and add the new server and private key

Host *name-of-ssh-host-here*

User *your-user-name-on-host*

HostName *host-fqdn-or-ip-goes-here*

IdentityFile *C:\\path\\to\\my\\id_rsa*

For the IdentityFile specify the path of your private key. For Windows make sure to use double slashes.