Using hooks in Joomla to add a Single Sign-On link in main page of Joomla

The easiest way to do what you are trying to accomplish is with template overrides, not plugin events (we don't call them "hooks" in Joomla). Ideally, you'd need to make a template override for com_users' login view and the mod_login module to make sure that both places where a user can log in are covered.

The reason I'm not recommending a plugin event is that Joomla 3 does not yet have a plugin event to add additional login buttons / links to the login page. This feature was added in Joomla 4 with my contribution of the WebAuthn plugin. The idea behind adding those events was your use case: Single Sign On implementations. But that's besides the point since Joomla 4 is still in beta.

As a side note, I wouldn't modify Joomla's database tables. It's not necessary. You can store additional per-user information in the #__user_profiles table. This is a more forward-looking approach, meaning that upgrading your site to Joomla 4 in a few months won't break your SSO implementation.