how can I bypass new login method security

Yes but it the bypassing the security and how did they get the passcode number and enter it at same time

No you have to find your own method no one can tell you how to bypass the restrictions but you cannot migrate to Jamaica without having the same issue over again

TOU BYPASSED, TIME TO GO SEE IF I CAN PROMOTE! ALMOST THERE!

Will post my method in a bit!

For the sake of account security, I changed the account login and after that I inadvertently changed the gamertag to another one and now when I start the games I get an error.

What do you mean you "inadvertently changed the Gamertag"? Did you by accident create a new Microsoft Account instead of changing the sign in alias of your existing account? Because changing the Gamertag can only be done on a specific website or on the console
and will cost you the equivalent of about 10 US$.

As for account security: The only way to make your account secure is by enabling Two Step Verification on

https://account.live.com/Proofs/Manage.

Changing a login name is simply "Security through obscurity", and is not a method to make something secure.

Surely you can come up with a way to bypass any security features.