You don't create a "second handle".
Just ensure you are logged out of any Twitter accounts, then just create a new account with a new name.
You will probably need to sign up for the new account with a different e-mail address to the one used for the original account.
There's no connection between the accounts.
I have a personal Twitter account, a Twitter account for @EMDiscussions, and a Twitter account for a hobby website I'm running.
It's not rocket science.
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