Cloning a sequence will only copy the steps and the settings, and if you plan to send the cloned sequence with a new email, you can change that in the sequence settings.
You will still have to enroll the contacts you'd like to have enrolled, either manually or via a SmartList (refer to the section 'Using a SmartList for Auto-Enrollment in a Sequence' in the article).
If you are looking for the cloned sequence to be an addition to the previous sequence, but want them to go into a new sequence at a certain point, you can use the Sequence Journeys function (new Sequence Enrollment) as a step inside of the original sequence if you still have the original sequence running.